Hiya! I'm working on my postgrad on popular music so i'll try and give you a rundown of all the factors from each person's perspective as well as in-general band "stuff"! This will mostly be from after 1967, as thats where the cracks started to show. If you want to see examples of anything i'm talking about, watch the film Let It Be!

Paul

The most factors (in my opinion) came from Paul's end. Since the group's manager, Brian Epstein, had died, the group needed a manager for their finances - under their own control, Apple was losing money at an alarming rate. Paul wanted his wife's father Lee Eastman to represent them, whereas all of the other members wanted Allen Klein to manage them, and they won out. Paul was incredibly unhappy about this (Song's such as "You Never Give Me Your Money were written about it), and Klein turned out to be a very bad influence, and it drove a wedge between him and the other members.

Paul also was itching to return to touring, and it was becoming evident in his behaviour; He was becoming an autocrat amongst the band, insisting on having every element of his songs played to perfection, and often using session musicians when he deemed George or John as not up to the task. George Martin felt the tensions started in The White Album and just got worse when the group were rehearsing in Twickenham. Paul's desire to play live again was not echoed by George or John (in fact, John only played live a handful of times after the group stopped touring), and eventually they compromised with the rooftop gig in 1969.

John

John is the most widely credited as the factor that "destroyed" the group, due to his relationship with artist Yoko Ono - he was the first to leave the group officially, whilst Ringo and George always returned. This is mostly an oversimplification, but its true Yoko was a catalyst for unhappiness - there had always been a rule of "no wives or girlfriends" at rehearsal or recording, and John frequently broke this rule by bringing Yoko along. She also overstepped boundaries, giving comments on other members writing and performances, something they found innappropriate. George can be seen complaining about Yoko on mulitple occassions on the film Let It Be. Yoko encouraged John's more experimental style, and their solo album Two Virgins was seen by the rest of the group as confusing and unnecessary. This led to John's overt dislike of Paul's songwriting, and he refused to perform on many of his songs.

Other factors were also in play - John became addicted to heroin, and it made him cold and indifferent. He wasn't interested in writing or performing, and when George threatened to leave the group, he simply suggested they replace him with Eric Clapton - he had been feeling George's guitarwork had been subpar in the recent sessions anyway. The others began resenting his songwriting and general apathy.

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George

George's motivations were slightly more straightforward - he felt he was being passed over in the group, with his writing overlooked and his parts often played by Paul (and sometimes John). George had spent time in Woodstock with The Band and envied their relationship, with each member contributing vocals and writing credits to songs, and his return to London just made the contrast even starker. This is fairly evident when we see that he only gets one or two songs per album, but was able to fill the triple-LP All Things Must Pass with songs immediately after the group finished in 1970. He was angry with domineering Paul and apathetic John, and felt increasingly bitter towards them both. Whilst he didn't officially leave the group, he stormed out on many sessions and had to be continually coaxed back.

He also unintentionally caused tension between himself and Lennon-McCartney - his new songs were better than they had ever been, and the pair knew it. They were nervous people would begin to prefer George's writing, and it made them defensive around him, alienating him further.

Ringo

You many have noticed that Ringo hasn't come up very much, and there's a reason for this - he really didn't get under people's skin. Ringo did leave the group briefly after constant critique on his drumming from Paul, but this was more like a sustained argument than a full departure. His drums were always solid, and since none of the other members could play near to his standard then he couldn't be overdubbed on records (The rumours that Paul re-did his parts is easily debunked). His alcoholism was a problem, but was overshadowed by John's drug abuse. Ringo often featured on the solo albums of his bandmates, and its often said that Ringo just liked hanging around with his friends and playing music. Again, watch him writing Octopuses Garden on the Let It Be film to see the joy he inspires in John and both Georges.

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IN CONCLUSION

Paul wanted to be in charge, John didn't care, George felt unappreciated and Ringo just wanted everyone to stop fighting.

This isn't everything, but its the majority. Cheers!

I think that the reasons why exactly the Beatles broke up are still to be definitively written down, specifically because Mark Lewisohn, who is the dean of Beatles historians, hasn't got up to that part of their story yet. What he has done, however, is teased the existence, in the media, of a recording of Lennon telling Ringo that they have plans for a further album after Abbey Road and a Christmas single, which, as Richard Williams in the Guardian says rather breathlessly:

Ringo Starr is in hospital, undergoing tests for an intestinal complaint. In his absence, John Lennon, Paul McCartney and George Harrison convene at Apple’s HQ in Savile Row [on September 8th, 1969]. John has brought a portable tape recorder. He puts it on the table, switches it on and says: “Ringo – you can’t be here, but this is so you can hear what we’re discussing.”

What they talk about is the plan to make another album – and perhaps a single for release in time for Christmas, a commercial strategy going back to the earliest days of Beatlemania. “It’s a revelation,” Lewisohn says. “The books have always told us that they knew Abbey Road was their last album and they wanted to go out on an artistic high. But no – they’re discussing the next album. And you think that John is the one who wanted to break them up but, when you hear this, he isn’t. Doesn’t that rewrite pretty much everything we thought we knew?”

There's a bit of salesmanship going on here on Lewisohn's behalf, of course, but it's fair to say that there isn't one specific reason why the Beatles broke up, beyond nobody really being that satisfied with being in the Beatles. If you look at the Beatles' official Anthology book, the reasons stated by the participants include:

Ringo Starr:

After the Plastic Ono Band's debut in Toronto [on September 13th, 1969], we had a meeting in Savile Row where John finally brought it to its head. He said, 'Well, that's it, lads. Let's end it'. And we all said 'yes'. And though I said 'yes' because it was ending (and you can't keep it together anyway, if this is what the attitude is) I don't know if I would have said, 'End it.' I probably would have lingered another couple of years.

John Lennon:

I knew before we went to Toronto. I told Allen [Klein, the Beatles' manager since January 1969...well, at least John, George and Ringo's manager; Paul McCartney felt Klein was bad news, and retained his father-in-law as his legal representative] I was leaving...I hadn't decided how to do it - to have a permanent new group or what?...Allen had said, 'Well, cool it, cool it,' because there was a lot to do business-wise, and it would not have been suitable at the time. Then we were discussing something in the office with Paul and Paul said something or other, to do something, and I kept saying 'no, no, no' to everything he said. So it came to a point I had to say something, of course. Paul said, 'what do you mean?' I said, 'I mean the group is over. I'm leaving!'

... I started the band. I disbanded it. It's as simple as that. My life with the Beatles had become a trap. A tape loop....when I finally had the guts to tell the other three that I, quote, wanted a divorce, unquote, they knew it was for real - unlike Ringo and George's previous threats to leave. I must say I felt guilty springing it on them with such short notice.

George:

I don't remember John saying he wanted to break up the Beatles. I don't remember where I heard it. Everybody had tried to leave, so it was nothing new. Everybody was leaving for years.

The Beatles had started out being something that gave us a vehicle to be able to do so much when we were younger, but it had now got to a point where it was stifling us. There was too much restriction. It had to self-destruct, and I wasn't feeling bad about anybody wanting to leave, because I wanted out myself. I could see a much better time ahead being by myself, away from the band. It had ceased to be fun, and it was time to get away from it. It was like a straitjacket.

Paul:

I must admit we'd known it was coming at some point because of his intense involvement with Yoko. John needed to give space to his and Yoko's thing. Someone like John would want to end The Beatles period and start the Yoko period, and we wouldn't like either to interfere with the other...

But Paul, of course, was the Beatle who didn't want to break up the Beatles:

I spent a lot of time up in Scotland where I have a farm. I normally go for holidays, but I began what was to be a whole year up there, just trying to figure out what I was going to do, and that was probably when it was most upsetting. I really got the feeling of being redundant. People say, 'but you still had your money, it wasn't exactly redundancy. It's not like a miner who's laid off.' But to me it was. Because it wasn't about money, it was about self-worth. I just suddenly felt I wasn't worth anything if I wasn't in the Beatles.

It was a pretty good job to have lost - The Beatles. My whole life since I'd been seventeen had been wrapped up in it, so it was quite a shock. I took to my bed, didn't bother shaving much, did a lot of drinking...

So, from that point of view, the decision to break up in September 1969 was John's, but it was a relief to George, and Ringo wouldn't have kept it going much longer. Paul took it hard, and exited to Scotland. In the meantime, Allen Klein told them to keep quiet about it, and they decided to get Phil Spector in to work on the Get Back sessions that became Let It Be, with overdubs. So for all that they'd broken up in September 1969, they...didn't.

Paul:

For about three or four months, George, Ringo, and I rang each other to ask, 'well is this it then?' It wasn't that the record company had dumped us. It was still a case of: we might get back together again. Nobody quite knew if it was just one of John's little flngs, and that maybe he was going to feel the pinch in a week's time and say, 'I was only kidding,' I think John did kind of leave the door open. He'd said, 'I'm pretty much leaving the group, but...'

But ultimately, Allen Klein as the manager of John, Ringo and George drove Paul away from the other three.

Paul:

I'd fallen out with the other three at once over the Klein thing. I didn't want him representing me in any way...so it was three against one. Never mind three against one - it was me against the world! It was me against three hundred million as far as I was concerned. The way I saw it, I had to save The Beatles' fortune. All we'd ever earned was in that company - and I wasn't about to see it go.

John:

...we were naive enough to let people come between us and that's what happened. But it was happening anyway. I don't mean Yoko, I mean businessmen. It's like when people decide to get a divorce: quite often you decide amicably, but then when you get your lawyers and they say, 'Don't talk to the other party unless there's a lawyer present,' that's when the drift really starts happening....it always gets nasty because you're never allowed to speak your own mind. You have to talk in double Dutch, you have to spend all your time with a lawyer, you get frustrated and you end up saying and doing things you wouldn't do under normal circumstances.

Paul made the break-up final in April 1970, releasing a press release about it to promote the release of his new solo album, McCartney, without consulting the others:

Q: "Are you planning a new album or single with the Beatles?"

PAUL: "No."

Q: "Is this album a rest away from the Beatles or the start of a solo career?"

PAUL: "Time will tell. Being a solo album means it's 'the start of a solo career...' and not being done with the Beatles means it's just a rest. So it's both."

Q: "Is your break with the Beatles temporary or permanent, due to personal differences or musical ones?"

PAUL: "Personal differences, business differences, musical differences, but most of all because I have a better time with my family. Temporary or permanent? I don't really know."

Q: "Do you foresee a time when Lennon-McCartney becomes an active songwriting partnership again?"

PAUL: "No."

Despite McCartney's equivocation - 'temporary or permanent? I don't really know' - the press certainly treated this as headline news that the Beatles breaking up.