A pair of muppets on Sesame Street who are always being accused of being gay lovers but are just really good friends. People seem to remember them taking baths/sleeping together, but I don't recall this (and I watched that show religiously as a kid). Their names were believed to have been taken from a pair of cops in the movie "It's a Wonderful Life," though apparently it's a rumor. Other rumors of Ernie dying of leukemia so as to stave off the homosexuality rumors are urban legends which get sheeple riled up. It seems that there's lots of rumors about them. Maybe because they're such lovable, well-known figures - people feel the need to tear them down, regardless of how they once felt for them.
Accipiter says Hi. Bert and Ernie *did* sleep together (Ernie ate cookies in bed, and the crumbs pissed Bert off.), and occasionally bathed together. Most of the time, it was Ernie with his Rubber Ducky, though.

you said "I never saw them do that, and it disagrees with my thesis, so I'm ignoring you. :)" to Accipiter

I remember Bert getting pissed at Ernie for eating cookies in bed, but it was totally different than that - they were NOT in bed together. They shared the same bedroom, but not the same bed. Ernie would eat cookies in his bed. Bert would tell Ernie, "You shouldn't do that. You'll get crumbs in your bed." So Ernie would respond, "Gee, you're right, Bert," and then go into Bert's bed to eat cookies.

That is a totally different thing. And I remember it like it was yesterday.

And I still haven't had any sort of evidence that they ever bathed together.