Too much of any nutrient will make you gain fat, but I wouldn't worry about protein. Its hard to eat alot of it but it is easy to go overboard with the carbs and fat though, so I'd pay more attention to that.
The best way to figure out how much protein you should have at each meal is by adding up your protein reguirement (at least 1 gram per lb bodyweight) then dividing it up equally over all your meals. So if you need 200 grams, and are eating 6 times a day, then you should get a little over 30 grams at a time.
I don't think there is a limit to how much protein the body can digest. Their was a myth floating around that you can only take in 30-40 grams at time. But what would happen to the rest of the protein in that case? A steak can easily have over 100 grams of protein, and I seriously doubt the body would only absorb 30 grams of it.