Bert and Ernie's ages have never been officially stated, but I consider them adults for these reasons:
1. They live in an inner-city brownstone apartment without a parent or guardian.
2. Most of their interactions has to do with adult characters (both muppet and human) rather than the child characters.
3. Their baby relatives (Ernestine and Brad) are shown as being MUCH smaller than E and B themselves. The size difference seems to be similar to an adult and a baby rather than a pre-schooler and a baby.
4. They drive cars and fly planes in some skits. Have you ever heard of a kid with a driver's or pilot's liscence?
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Bert and Ernie are adults. They're best friends who live like roomates. In fact, the only kid Muppet characters are the forever 6-8 Big Bird (despite Carrol Spinney himself being 70+), the forever 3-7 Elmo (as Kevin Clash himself said in his "My Life As a Furry Red Monster" biography), Zoe (who was intended to be basically a female Elmo [hence why they're best friends]), and newcomer Abby Cadabby (the little fairy godchild).
All the remaining Muppet characters--like their other family members in The Classic Muppets--are adults (especially Oscar) or teens (Telly Monster and "Baby" Bear [he's not the true baby of the family anymore, what with his sister, Curly Bear around]).
Actually, Kermit the Frog is a Sesame Street character,
Wrong. Yes, he did appear in the old school episodes of the show, but he is nowadays nothing but considered a "guest star" when his classic skits resurface and he himself is a timeless adult (despite the on-again-off-again possible canonity of Muppet Babies, what with it being merely a dream sequence in Muppets Take Manhattan or reality via home movies in Muppet Family Christmas [the latter brought the entire Muppet family together, although the babies themselves were merely in home movie sequences]).
In the words of Baby Animal, it's time for this topic to "GO Bye-Bye...Yababababa":