Okay, to get one thing out of the way: I personally like Cut Man, and he's one of my favorite Robot Masters, if only for the fact that he was the first one I ever beat in a Mega Man game. His depiction in Mega Man Powered Up and the Mega Man Megamix manga have only further endeared him to me. With that said... well, let's face it: in the old Ruby-Spears Mega Man cartoon, DLN-003 was seldom able to "make the cut." YouTube user CobraRNZ illustrates this through an amusing montage of Mega Man's foe continuously being cut down down to size:
This was fun to watch, but was sadly missing one of the best moments from the series' entire run, which also happened to be a "Moment of Awesome for Mega Man X. I'd link to it here, but it seems there is only one upload of the episode on YouTube, and that lacks the moment in question (which could be why it's missing. Sad, really).
Source: Attack of the Show
Edit: Reader Wingus has tipped us off to a video of his own making, giving one of Cut Man's misadventures that good ol' "Yakkety Sax." Check it out: