Hi, I just wanted to let you all know what is going on, since you were all so nice to offer suggestions! I called the vet, who wanted me to bring Bullwinkle in. He had his blood and his urine tested. The results were that he does not have diabetes, and that his kidneys are functioning fine. She didn't see anything else funny with the blood or the urine.
So basically, she thinks that it is a behavioral thing, she called it psychogenic polydipsia, where he's drinking more because he's bored. This still seems strange to me, first, because I don't see how he could possibly be bored with all his out-of-cage time, and all his toys. It also seems strange that it seemed to start suddenly one day.
But she wants me to measure how much he drinks in 24 hours for the next few days, to see if he's drinking too much for a rat. I measured today and he drank 50ml. I measured what the two rats I have in a different cage drank, and they drank 50ml between the two of them, so it's like he's drinking twice as much. We'll see what the vet thinks, I will let you know! Thanks again