Um, they didn't? The third century crisis is a great example. Gaul and Britain were operating independently from Rome, as were the eastern provinces, centered around Palmyra. In this period there were something like 20 emperors in 50 years who came up through the army. The place, at the time, was a mess.

No expert by any stretch but suggestions I've heard... During the Julio-Claudian era legions swore direct loyalty to the Princep. Even the most powerful provincial governer with the best troops couldn't match the numbers under direct Imperial control. Plus as equals under an accepted Imperial dynasty regional commanders were apparently pretty accepting of the status quo. Ambition would manifest in vying either for profitable postings, in competition with peers, military honour or fleecing locals. None of which are focuses which allow much scope for forging Empire wide alliances and conspiracies. And if you start plotting to Empire build why would your fellow Generals not reap the rewards of ratting you out or just decide on claims of their own? And under the Julio-Claudians leadership had legitimacy too. Caesar and Augustus's names had reach, even through the likes of Caligula. The only other popular appeal to 'right' would have been a claim to support Senate Republicanism which no one really wanted to fight for. Being a popular General on the Rhine or in Iberia wasn't reason enough to rule.

At a guess wealth was a factor too. Prior to Nero there was a load of it to be rinsed out of local corruption. So why fight if everyone was getting rich including the Empire itself? Even the poor in Rome had free grain from Egypt, another resource under the Princeps control and another thing some fella up in Gaul with a couple of legions couldn't offer the masses in a revolt.

All ended with Nero and the dynasties death though, as others have said after that stability was far rarer. No solid Julian familial claim to assert authority and in the year of the Four Emperors it was proven that personal ambition could reach the height of Princeps. Anyone could be legitimate with a big enough army.

I blame Nero really.

That's the thing. Rome delegated so much power to it's distant provincial governors that they had a nasty tendency to split off and try to seize power. The Crisis of the Third Century, as noted by another commenter, is a great example of this. The Year of the Four Emperors is somewhat a good example, as are the Civil Wars of the Tetrarchy.