As far as I remember from my schooldays, seawater evaporates to form clouds and rain falls from clouds back onto the land and sea. The rain that falls on the land finds it way back to the sea eventually. I think that's the gist of it anyhow.
I appreciate what you say in your last sentence, but I can't help feeling that the melting of icecaps and glaciers must be raising the sealevel. Just look at the size of those chunks of ice that hit that cruise-liner last week - this ice has to melt and have an effect on the overall sea level the world over, and this apparently is happening daily according to the global warming theorists.
It appears to follow that if there is a greater volume of water in the sea, there is more to evaporate to form clouds. Ergo, with more water in the clouds, there is more rain to fall on the land causing these floods in Bangladesh and Tewkesbury if it comes to that!
So I suppose no, it's not possible to have more water on the planet than already exists if we are speaking of the sum of liquid water, ice and water vapour combined. However, if you're thinking of liquid water alone, yes it can be increased by the melting of glaciers and ice caps.
I await correction with trepidition!