Are Bert and Ernie named after characters from It's a Wonderful Life?

It's been stated in numerous articles, websites and television shows that Bert and Ernie were named after the policeman and cab driver from the film It's a Wonderful Life, who were also named Bert and Ernie.

There has never been any official confirmation of this statement, and most sources from the Henson company have said it to be merely a coincidence.

For instance, the Henson.com website used to have an "Ask Henson" feature, whose FAQ contained the following item (you can still find it via the Internet Archive, e.g., at http://web.archive.org/web/20000303115742/www.henson.com/ask/index.html):

"My husband and I have a bet. He says Bert and Ernie are named after characters in It's a Wonderful Life. I say no way. If I win he cleans the house for a month. Who's right?
Put your feet up and turn on the television! While Bert and Ernie's names have often been linked to those of the cop and cab driver in It's A Wonderful Life, Jim Henson and others have always said that that is just a coincidence. Apparently, Bert's and Ernie's names were picked just because the creators/writers liked them, they fit the characters, and they sounded good together."