I've always thought twice before doing so, but I am not so rigid that I would not. I do find, however, that often the "And" may be removed and still have the meaning, intent and grace of the writing remain intact, in which case it is not necessary.

"But this rule has been ridiculed by grammarians for decades,"

I realize he was trying to make a point (and a funny), but I think the "But" can be replaced with "Yet" and not break any rules (or is there a rule against "yet"?) whilst retaining all the meaning. Good editing requires the removal of the unnecessary, the redundant and the superfluous. ;0)