Drains for washing machines are supposed to be two inch pipes.

if you cut a y in then you want to put a piece of pipe running up to so its a few inches below the washer, not down at the ground. So it seems to me more likely the drain is clogging up somewhere.

There's really no way to know without a bunch more info, which might make more sense for a plumber to collect.

If the laundry room is near a kitchen then it maybe the kitchen sink has clogged up the same line that the washing machine and laundry tub run into.

I would go to see if any of the other things, especially the kitchen sink seem to be draining slowly. I would also try to track the pipe down into the crawl space or basement and make sure it has enough fall/no dips or flat spots or doesn't have any obvious points where instructions for 90 degree turns that might be clog points.