If your Norton or indeed any anti virus is not on by default and switching it on doesn't help in that the next time you start up its switched off again then there is something to worry about.
Personally I would scan the PC with the anti virus in safe mode and run a couple of ad-ware programs and see if that finds anything. If not then I would delete and reinstall Norton although this may not be as easy as it sounds. Sometimes Norton comes off from add/remove programs ok but other times its so deeply inbedded you have to download a removal tool from Symantec to get rid of it.
Leave this link open for a while before removing Norton and see what suggestions others have. In the meantime do a full scan with anti virus and adaware or whatever else you have. Either something has got on your PC which is switching it off or the program has a fault. You could also download the free AVG and run that and see if it finds anything. There is no harm in doing this as long as you make sure it doesn't run by default at the same time as Norton as then they would conflict. And of course you can always remove it afterwards if you wish.
I'm afraid I'm not a Norton fan.