Declining third-party support. Sites with APIs typically develop a culture of third-party apps -- image uploaders for photo-hosting sites, for instance, or applications that integrate directly into the service, such as Facebook's massive roster of games. If development of such applications has fallen off, that could be a sign the service is losing its user base. If the pace slackens not because of market saturation (you can only have so many photo uploaders) but because of genuine programmer alienation -- to the point where word filters out into the general user community -- that's a bad sign.
Changes in terms of service or arbitrary behaviors. Many people leave a Web service behind not because the service itself is endangered, but because of things the service has done. A common reason for this is changes to the terms of service, which can spark a massive user backlash. It doesn't help that terms of service are all too often pools of mud, where the implications of any changes are unclear unless spelled out with total precision. Think of the recent flap over DropBox's clause indicating it would turn files over to the government if asked -- which forced the company to add wording to the effect that your stuff remains yours and they won't mess with it unless they have no other choice. (In its own words: "These Terms do not grant us any rights to your stuff or intellectual property except for the limited rights that are needed to run the Services.")
Different folks have different thresholds of tolerance for such things, so what ticks off your neighbor may not seem as egregious to you. But if you hear about such a thing happening with a service you use, pay attention, and give the ToS a fresh read whenever you're asked to reconfirm your acceptance.
Read the terms of service
Speaking of the terms of service, that's the one part of any service you shouldn't ignore, since it spells out what can and can't be done with your data. It doesn't help that most terms of service are terribly arcane, with crucial points buried within multiple clauses of pure lawyer-speak. Here are several major clauses that appear in a site's ToS, as they affect movement of user data.
Rules about third-party programs. Many sites explicitly disallow the use of unapproved applications designed to scrape or harvest site data, on pain of termination. If you're leaving anyway, this threat isn't quite as weighty, but it might cause trouble if you are relying on such a program to back up your data on a regular basis. These rules often cover the service's stance on data portability -- they may not come out and say that data can't be exported from the service, but they may add rules like this to make it massively inconvenient.