Can a cartridge be so dirty it stops the console from working? I was at my friends house and he wanted me to plug in majoras mask and it stopped working, but the red light comes on. It just doesn't play?
If some of that funk that is on the cart comes off on the pins of the system, sure it can make it not work.
Get you an old t-shirt, a credit card, and some alcohol. Put the t-shirt over the card, put some alcohol on it and stick in down in the N64. Clean it out that way the best you can. It may take a few times, but try to get the dirt out. Good luck.