Try redownloading them.I think in my case its because I bought a second 360 to use as a dev kit, but the new unit had hdmi. So I took the hdd off the old 360 and plopped it on the new 360 in the tv room. Hence why now if it doesn't connect to XBLive, then my purchased games run in demo mode. Not a huge deal, it only affects Robotron and Paperboy But still, it irks me when I can't play content that I've legally purchased.
What I have discovered is that the account holder that purchased the game can download and run it in "full" mode on any unit. However, other people can only access the fully purchased title on the unit that it was initially downloaded to. i.e. A user downloads on to XBOX a, anyone using XBOX a can play the full game; subsequently the original user further downloads a purchased title to XBOX b only that user can use it on XBOX b, other users will automatically be pushed into the demo version.