The only time I've had a problem with this is when I've created a panelized board myself - laid out a panel with a number of boards of the same or different design, In this case, overhanging silk screens end up on neighboring boards. Where the fab house has put together a panel, it's never been a problem.
What I like to do is put the overhanging part in a layer that doesn't show on a silkscreen. In Eagle, the document, tdocument and bdocument layers don't go into the silk layers on the Gerber files, so putting the overhanging parts in one of these layers works well. For example, with a right-angle BNC connector, the threaded part and barrel on on the document layer. You can tell the proper placement at the edge of the board, but there's no worry about the overhanging part as it won't be in the silk layer. In the picture below, the black lines are in the normal silk layr (tplace) and the magenta is in the tdocument layer.