I did a show once with an actor who decided to run two of his lines together, which meant that saying my line (which was only a single word: “No!”) muddied things a bit. He had an emotional reason for making this choice, even if it was a bit misplaced. I ended up cutting my line out, because the way he was handling the scene just made it seem messy and as if someone (me, probably) had screwed up a line somewhere.
I had plenty of lines in the show, so losing a word was hardly a problem. However, the lines were written as they were to produce a laugh, one we never got because of how the actor was playing the scene. All the more amusing, really, since the actor in question considered himself to be a comic. But I always felt badly about it, because it was the audience who lost out. (And no, the director did nothing to fix the problem. C’est la vie.)
Overlapping dialogue is fine, even when the playwright hasn’t written it to indicate overlapping. It makes things more realistic, and sometimes helps to convey urgency or passion on some level. It’s nothing that you want to do too frequently, because the other actors may start to feel that you are “all about you”, and not about the play. Nor do you want to overlap more than a word or two.
You also want to be sure that you aren’t overlapping any important information or emotion. Never step on another actor’s moment. And never kill a laugh intentionally, as my scene partner did. Audiences love to laugh; give them every opportunity!
Also, be careful in doing this when the line you’re overlapping belongs to someone who has a small part in the play. If it is really necessary and appropriate, that’s one thing, but do remember that actors with small parts relish every word and moment they get on stage. Let them have them! If you’re the lead and they have twelve lines, you won’t endear yourself to them by stepping on one of them.
If you find yourself talking at the same time someone else is speaking, go back and check the script. Make sure you’re handling your end properly and that you haven’t misunderstood the scene or memorized it incorrectly. If the problem turns out to be your scene partner, have a word with your director in private. With any luck, he’ll fix the matter, and if he doesn’t, do what I did in the above instance: make your best judgment about what you can do that will best serve the play in this instance.