Generally, a joke that needs to be explained falls into the 'fail' department. Unless the joke is that you want the audience to go, "Huh?"
Watch enough old dubs and you'll run into a situation where the English performance is basically a straight reading of the literal translated Japanese script. They usually aren't much better than the dubs where the director and actors take more liberties. Sometimes, especially for comedies, the English adaptation is funnier to me. Don't get me wrong, I'm not saying that all shows should be treated this way. But I enjoy the dubbed versions of Lupin III (Pioneer/Geneon) and Crayon Shin-chan as much as the subbed Japanese (for different reasons, of course).
Excel Saga has a few gags that are funny in a different way in the English version compared to the Japanese. One that I can remember is when Excel goes to America and she ends up looking like Sailor Moon. [spoiler]The joke in the Japanese version is that the Japanese actress playing Excel (Kotono Mitsuishi) also did the voice for Usagi (aka. Serena) in Sailor Moon. The dub makes a joke that the actress playing Excel (Larissa Wolcott) wanted to play the part of Sailor Moon (Usagi/Serena) but couldn't because it was being dubbed in Canada (and ADV is loacted in Texas, USA).[/spoiler]
ADV's Super Milk-chan got both treatments. The version that was played on Adult Swim was faithful to the original Japanese and while I enjoyed it (mostly), I can see how most English-speaking viewers would have turned it off. Now, according to
AnimeOnDVD's review, the US version comes with another disc with a drastically different adaptation that seems to be tailored to American tastes in comedy. I've never got around to renting Super Milk-chan and watching the other English dub/adaptation, but having seen the AS aired/faithful-to-the-original version, I suspect that taking some liberties with the script of this show would have made it at least more popular on US TV. Heck, Shin-chan got picked up for another season (and some of the changes for the English version are just plain WRONG, but in a good way).
If you have enough knowledge of Japanese culture you can probably get as much amusement (if not more) by watching the subbed version and getting the original jokes (plus the original vocal performances). I watched Lucky Star in Japanese without and with subtitles and even with my otaku knowledge, didn't get the full impact of a joke sometimes.