No, not really (not if we're talking about humans, and as seen in mine-craft anyway).
The fastest any human has ever swum is 2.29 m/s, this record was set by Micheal Phelps who is about as swole as any human could ever hope to be.
The terminal velocity of a raindrop is about 10 m/s. Even ignoring a whole host of factors making this nigh impossible, the water coming down is at least four times faster than you could ever hope to swim.
But lets talk about those factors.
First off you'd also need to push your own weight up (Phelps' record was completely horizontal remember), this is something most people can do simply by climbing a ladder so isn't actually a hard part.
Secondly the water coming down probably isn't a continuous volume of laminar-flow water, It's just a bunch of drops (if it is a continuous volume its probably travelling much much faster than raindrops anyway so you've still got no chance). This means that the reaction mass for each of your strokes is much lower so you get insanely low efficiency while swimming. To simulate grab a swivel chair, take it outside and "swim in the air" till you move. Now have a friend pour a sprinkler over you while you do it. You will notice both times that you don't really go anywhere.
Thirdly you'd also need incredibly coordination and balance even if you could provide the upthrust.
Your best bet is probably to try this in a gravity-less environment such as a space station.