Typically not, unless they manage to get really lucky and find their script in a bidding war the first time out the gate. Or unless they get a script actually made into a movie (very rare, even for successfully beginning screenwriters), and the movie does well. Once you start hitting it big in screenwriting, you can make a very nice living. But most writers go through a very long dues-paying period, and that's not including the ones who never even make it to the starting gate.

Now, some people might be tempted to say "But if I can make a couple hundred thousand from a spec sale, can't I just write a ton of specs and get rich?" Probably not. The way to become known amongst studios is to take a bunch of rewriting or project-based jobs, preferably after having sold a debut spec. If you figure that the average spec could take anywhere from 6 months to a year to write, and that most specs don't sell, you're looking at a very uncertain income stream. And even if you do make a nice sale, you have to factor in everybody's cut: agency, manager (if you have one), lawyer, taxes, etc.

In general, screenwriting follows the pay function common to most of Hollywood: you spend a long time making close to nothing, but if you ever manage to make it to the higher echelons, you can make a killing. Skill plays a role in this journey, but luck is arguably even more important. Understandably, not everyone has the stomach for it. If you want to get rich, there are far easier and better paths to doing so.

Writing is really a game you have to be in because you love it so much that you can't imagine doing anything else. It can be a fun lifestyle if you manage to make ends meet, and a great lifestyle if you manage to become successful at it.