Is it possible for an adult beginner to get really good at the fiddle?

Yes, absolutely! I'm the proof. I started playing the fiddle when I was 30, and now I play professionally and teach.

However, "good" is a relative and abstract word. When you say "good," are you comparing yourself to professional players? If the answer is yes, stop doing that! It was never my intention to play professionally. I just liked playing and did it a lot. I enjoyed learning tunes and techniques. After a while, I got hired to play. This isn't unusual, but it's not something you can expect. 

There's a prevailing myth that in order to get any good at anything you have to start as a child. There's a grain of truth here. If you plan on being a concert violinist, it's best if you started young, because it takes a long time and it's a lot of work.

If, however, "good" means good enough to enjoy playing (and have others enjoy it, too) then it doesn't take very long at all. Some take to it right away. The first several weeks are the rockiest because it's not a natural thing to learn and you have to be willing to sound terrible for a time. It also takes dedication and discipline, qualities that adults possess more than children do.

Think of learning an instrument as a journey and not a destination. Even the most accomplished player knows that they haven't yet arrived at being the player they'd like to be. You can have enjoyment at every level. Enjoy being a beginner! When you draw the bow across the string and you get a nice full tone, that's no small achievement. And when you can play a jig, and it gets someone to tap their foot, that's no small achievement either.

It's a fun and rewarding journey. If you haven't yet embarked, perhaps it's time to do so. If you are already on your way, enjoy the trip!