You can romance all of the companions if you want. However, if you romance Piper, and then choose a 'flirt' dialog option with Cait or someone else in the presence of Piper, she'll dislike it. I believe the same can be said for all of the companions.
If you talk to the singer/"lady of the evening" at the Third Rail in Goodneighbor and choose the 'flirt' option while talking to her, your companion will dislike it.
So as long as you don't let your romantic companions see you romancing your other romantic companions, you'll be fine. I've already romanced Piper and Cait, and now I'm working on romancing Curie. I'm a straight dude, so I'm still debating pursuing the romance options with the male companions. But in the interest of completeness and getting all of the extra perks, I probably will anyway.