Of course. DC successfully sued Fox Comics over Wonderman back in 1939 or 1940. DC also sued Fawcett. That was eventually settled out of court with Fawcett giving up Captain Marvel.
For clarity, I should mention that DC got an injunction against Fox's continuing Wonder Man. As I recall, they didn't go through a full copyright infringement suit with damages. There may have also been an injunction against Fawcett for Master Man.
Likewise, the Fawcett story (which was an infringement battle) is much more complex. Fawcett originally won the suit, claiming that DC had abandoned the copyright when licensing it for the newspaper strip that didn't usually carry a copyright statement. In other words, they argued that DC shouldn't have standing because they didn't require their contractor to protect the property.
On appeal, that decision was thrown out, because (as we've said a lot, here) there ain't no such thing as abandoning a copyright. So Fawcett lost the case, but the two companies agreed to settle the payment amount out of court, rather than paying lawyers to review every panel each character ever appeared in. (I point that out, because "settled out of court" gives the impression that the case was left undecided, whereas DC definitely won on the appeal.)
The general claim is that they (Fawcett) stopped publishing comics more because the profits weren't there anymore, rather than because it was part of the settlement. I don't know if that's ever been investigated from reviewing the contracts, though.
But regardless, the upshot is that, yes, comic companies sued one another on occasion. However, the key is probably that you need to know about the infringement, care, and have the money to pursue it. Would anybody at Harvey be reading Marvel books and remembering an obscure character from fifteen years ago? Would they have the money in the budget to sue? Those are the main factors, usually.