Yes. Listening to music, trying to play over it. Get a metronome, play with it, even if it's only stretching exercises. You'll get a feel for it. Try playing the same song slow or fast, have someone with a sense of rhythm tell you how well you're doing (or play over a song that you can control the speed of -- midi software is great for that).

Then there's some music theory you can look for: Pop and rock music will usually have the base of the chord (unless it's a reverse chord) on the first time, with lots of emphasis on it. The third time will have some emphasis, but not as much. Reggae music will have it in a different way, with more focus on some upbeat times.

You have to practice, but you must practice smart. A very good exercise I had was playing in larger ensembles playing anything from classical, blues and some jazz music. I would sit next to a very good tuba player (with something like 50 years of experience) who I could listen to when music notation got problematic. With time, your brain starts to understand things and it helps, but it takes a lot of time and effort.

I'll add that even your ear can develop and recognize notes with more ease. In my case it's not the absolute hearing (finding out what note some random sound is), but rather relative hearing (telling the distance between two notes). I've been able to develop this over theoretical exercises where a teacher gives you a simple song and you have to write it down. Go from simple to complex, by removing and adding notes. Study scales, chords, how they sound and what they can evoke. With enough practice/training, your brain will absorb it and won't be entirely incompetent.

After 1 year of casual music training, I completely froze up on stage, losing my sense of rhythm and being unable to read the sheet I had in front of me. That's when I enrolled in the big band-like ensemble I menationed earlier -- suddenly I almost doubled my practice time, played genres I didn't know very well or listen to in my free time, and one year later, I was a good second bass player. After the third year, I could play on my own with very little problems. The fourth years was spent perfecting my technique and I spent the fifth year getting to learn how to improvise. It's been about 2 years since I stopped taking classes and playing seriously, but the sense of rhythm is still there.

It takes a lot of effort and perseverance. Things seem to move in staircase way: you start with a wall to get over and it seems like you'll never do it. At some point it clicks and you can enjoy yourself. You're now on a plateau of comfort, until you hit the next wall, and the one after that, and so on. Some are way harder than others, but they're all normal.

And for the love of God, please don't compare yourself to others (you won't be able to keep yourself from doing it). I used to do it all the time. Comparing myself with players who were half my age and twice better, regretting I didn't start earlier or thinking 'what is it worth? they're still better anyway'. It's not a contest, but you have to remember that mastery comes with time and rather compare how many hours of very serious practice you've done compared to others.