I know on my older jailbroken Kindles, the jailbreak is very fragile and they can no longer be connected to the internet, partly because of Amazon's agressive OTAs and partly because Amazon disables developer mode and hence the jailbreak every time my Kindle 4s touch the internet.
(PS. If anyone can suggest a way to prevent this and secure my jailbreaks, it would be much appreciated - I've tried 3 different OTA blockers and none work reliably)