Shakespeare is often credited with coining a number of words and phrases now commonplace in modern English. Were there any created terms of Shakespeare's that didn't catch on? If so, do we know why some did and some didn't?

Ok, I think I can provide some insight into this.

My main source is David Crystal, "Think on my words: Exploring Shakespeare's Language", which you should really take a look at if the topic interests you; Crystal is also quite fun to read, so.

There are some things that should be clarified about the idea of Shakespeare inventing words (as other users have pointed out).

Many of the words he "invented" are not actually invented, as in, they're not actually neologisms, but rather creative usages of already existing words. What I mean by that is, our favorite bard liked to use tricks such as derivation (a new word is created by adding a suffix or prefix to a preexisting root) and functional shift (usually, a noun is used as a verb, very common even in modern English) and sometimes he simply modified a perfectly known word in order to make it fit into the meter (e.g vast becomes vastie, a nice two-syllable word, handy when you're trying to write in iambic pentameter; for the record, this in itself formally classifies as derivation, the difference being that there's no subsequent change in meaning).
He is credited with more words than he actually invented. There are two reasons for that. One, there are some words attributed to him simply because he was the first to write them down, when it's reasonable to think that their usage was established well before that (Crystal mentions 'sblood!, "God's blood!", as being first recorded in Henry IV, when it's likely that Shakey took that right out of London streets). That's also common with idioms and proverbial expressions (e.g brevity is the soul of the wit), which he merely made more widespread thanks to the sheer popularity of his plays at the time. Two, everybody loves Shakespeare (rightfully so, if I may), and what that means is that Shakespearian texts have been studied more accurately and by more people than most of his contemporaries'. Which results in several words being "first" recorded in his plays when in reality there are earlier texts already using them (as it has emerged overtime).
Him inventing words was really nothing too special. Shakespeare lived in a time in which the English language was trying to better define its own identity and scope (the first attempt at an English dictionary, A Table Alphabeticall, appears precisely around this time, in 1604), and literate people were inventing words all over in order to try and elevate the language, making it richer and suitable not just for everyday usage, but also for higher registers (such as literature, of course).

So, with all that in mind, we can say that Shakespeare "invented" about 1,700 words. To answer your first question, yes, of course not all these inventions caught on; in fact, only half of them continued to be used, which is still nothing to laugh at. As to why these particular words lived on, in some cases they did because they were already in use (see second point). But aside from that, there probably isn't a definitive answer to that question, as it leans into the realm of why languages evolve the way they do at all; i.e there is no real logic in the way certain morphemes / words / idiomatic phrases fall out of use or survive through time.