Yes - and I have the recordings to prove it! (of course, assuming there aren't medical or physiological conditions that would prevent it)
I think the "real" question is how badly do you want to sing well? It can take a lot of dedication, effort, and persistence under the guidance of a good teacher. You won't hear drastic differences overnight, and there will be plateau's and maybe even times you feel like you're going backwards. But in the long run, the results will show if you put in focused practice and keep up a can-do attitude. It's like gymnastics or any other sport; there are a lot of muscles to train and coordinate (even if you can't see them), and you won't get to the Olympics overnight!
To give you an idea: I started out with *maybe* 3-4 notes that I could sing in tune with the piano if I tried really hard, and none of them sounded good or were reliable. Now I have almost 2 octaves I can reliably vocalize on. Some notes still don't sound great, but others are sounding quite nice and I even sometimes have things like vibrato coming in. I've practised almost every day for several months, and still have a long way to go.
If you enjoy singing in the way that sbhoa said, or are motivated enough to achieve the result, lessons are worthwhile and practice is fun. If not, it will be a chore.