This is a topic of hot debate.
Personally I side with "Ernie and Bert". Originally in the sixties and seventies, it was always "Ernie and Bert". Their first record in 1971 was "Havin' Fun With Ernie and Bert". As Bert's personality developed "Bert and Ernie" began to appear more often like on the 1975 "Bert and Ernie Singalong" album.