There is no point running both and Norton will probably conflict. Personally I would just run AVG, it is free and I've put it on a lot of computers in the last year. Most viruses are annoying but will not cause untold damage. They are easy to get rid of and AVG is as good as any.
Some Anti virus progs will coexist reasonably well, others will create havoc and can clash and badly destabilise the system. In short not a good idea as the potential consequences far outweigh any benefits you may get from two tier cover.
If you want extra confidence you could use :- Microworlds Free eScan Antivirus toolkit click here It's an on demand scanner only so shouldn't cause problems.You can always use one of the online checks like Panda click here