Can you get diarrhea by drinking too much water?

Generally no. The body has mechanisms to get rid of the excess water, but under certain conditions - if it is taken in very rapidly or in certain disease states, it can cause water intoxication - diluting the blood so that sodium levels can get very low - if low enough, it can cause seizures, coma and death. Diarrhea rarely occurs, because the water is absorbed in stomach and upper intestine, but it can.
No. Watery is generally well absobed. If you had water toxicity, drank so much water you lowered serum your sodium to a dangerous level, you potentially could have diarrhea but this too would be unusual.