My uninformed guess is that it has to do with look ahead. If you hard code the 'e' it behaves as it should:

$word=~/(e)([^e])e/;
[download]

My understanding was that you could not use a reference to the matched result until after the pattern has matched.

I eagerly await the real answer from those who know

update: didn't have to wait long for that one. Should have practiced some look behind ;)

update2: As usual, I've confused my terminology. In studying japhy's answer it looks like I had the right idea, but really what it comes down to is that ([^\1]) is the wrong way to do a negative lookahead Pg. 203 of the Camel Book (3rd edition) offers a good explanation of this.