Well, this may or may not be true but I think you may be confusing having a suit filed against you and having judgment rendered against you. To that point: I would say it depends on who's song is being covered and by whom. If an "A list" band covers another "A list" band's song I would contend that the attorneys (given their nature) for the original song writer, publisher, and or artist (which could all be different people/entities) will sue everyone involved (just for affect) and let the court settle it. Now if some guy/band in a garage releases the same cover in low distribution it is less likely (key word being less) that mass suits would be filed. However, more likely is the band gets sued then turns around and sues the recording studio claiming that they didn't know and the studio should have told them this was a problem which is why in my original response I suggested indemnification form the band and or 3rd party suits.
Lots of what ifs I know but not out of the realm of possibilities, I've seen suits filed for a lot less . At the end of the day the only liabilities would be for royalty payments (not likely to amount to much if low distribution) and legal fees which could be more significant. Not sure where you're located but here in NJ I suits get filed for a lot less....I need to move