Is it possible to do too much networking? Sure, you can overdo anything, but if done right, there's really no such thing as too much networking.
Question is, how do you do a heavy networking campaign right?
Be everywhere: If you're going to be part of a group such as a chamber ribbon cutting team, go to all their events. You'll be an expected fixture at that particular type of gathering, and missed when you're gone. So you're "present" even if you're not there.
Be selective: Sounds contrary to the first item, doesn't it? But if you're going to be a member of a lot of groups, be as diverse as possible. If half the people in your book club are also in your leads club, you're not getting exposure to as many different people.
Be impeccable: Have great manners or you'll be the boorish guy or girl who goes to everything, hands over their card to unsuspecting guests, and then hits the buffet. You want to be a networker, not a freeloader. Make sure that you are an appropriate addition to the group. Don't join just to sift through for business - try to have other reasons to be there. This is especially important when people start to notice: "Oh, you're just everywhere, aren't you?" Have a light or humorous response with a good reason besides peel and eat shrimp to be there.
Be focused: You need to know why you're out there and what you want to do. For example, you have a new business and you need to get the name out there. Or you have a new job and you need to generate a lot of new leads right away.
Be realistic: Be honest with yourself: are you getting results from your networking or are you just trying to avoid the stuff you don't enjoy? If you're an extrovert, sometimes you end up networking just for the fun of it, not working at making new relationships or strengthening old ones.
Be done with it: It's like a political campaign. At some point, you'll reach a saturation point. You can't maintain a high level of intensity forever. After a while you might get tired of the pace and start going through the motions as opposed to really working to meet people and develop relationships. Taper off, don't drop out of everything at once. You'll need to attend those events at a certain level to maintain your new relationships. And, you need to do all the work that your networking campaign has generated.