I have an induced gas furnace that's about 15yrs old. I'm in Texas. The furnace will run fine for quite a while trying to reach temp but before it gets there it will shut down. It doesn't shut down completely though. The draft inducer motor continues to run and it will restart within a minute (the pilot goes out and relights). When its very cold and running for a long time it will do this several times before reaching the desired temp. When it doesn't need to run very long to get to temp it doesn't happen. It seems to me like the gas flow may be stopping causing the pilot to go out. I tried the following:

New T-stat
Clean filter
Clean furnace/burner area.

Any idea what this might be?