Yes you CAN. That doesn't mean it is not a hell of a lot easier to get the same results by using other tools though.
If you want to get fit (build muscle), and you're not using a barbell, you're most likely making things harder for yourself. If that is what you want to do, fine, but recognize that you're taking a more difficult route.
Using weights indeed allows you to fine-tune your workouts more. Want to do 8 reps instead of 5? Just take some weight off. This is indeed harder to finetune with bodyweight progression.
If you talk about bodybuilding, max strength gains, etc, I agree you need to use weights for the fastest/best results.
But if your goal is to "get fit", my personal opinion (or preference) is that there is nothing better than bodyweight. Different muscle groups working together, balance, etc.
I guess it all comes down to personal preference. I got bored lifting weights, but absolutely love those bodyweight trainings. All I'm saying is that it's a viable option where you can build a lot of strength. Is it optimal for size, strength or endurance? Probably not. But it's not something to dismiss either.
Your body needs time to adapt to exercise, and body-weight exercises can provide a longterm plan to get that growth without threatening some serious injuries.
Also, that's a false comparison. Nobody said that you need to get off the couch and go straight to squat 500lbs to be in shape. You need a proper training plan with reasonable progression, and you need to start as low and as light as necessary, no matter the training stimulus (BW, barbells, dumbbell, kettlebells, bands, ...). The difference is that weights can provide an external force that can be scaled to your strength in many ways that BW training cannot, because in BW training the only weight is your body and you need to use leverages to modify your training stimulus.
Finally, BW versus weights is a dumb debate. They complement each other, and a well-rounded athlete should do both.
*to be fair, I'd recommend that a new trainee start with anything but barbell stuff. Simple bodyweight training 3-4 times a week, playing a real sport once-twice a week and doing some form of cardio twice a week should be the starting point of any athletic career, at least for 3-6 years. I started training with people who had an athletic past like this and they blew past me at an incredible speed progress-wise, and ended up stronger, healthier and in better shape than I ever was.
Fast forward a decade, and my right shoulder still remembers these pains - I dunno if the weights caused it or some previous injury, but when I started training with my own weight, one sign that I was getting ahead of myself was that small pain. Now I can progress slower, but still I have to be very careful. Some of the exercises with heavy weights are just not natural for the body and simply exhaust/strain it more than it should. If you are experienced, you can work around these limitations, but I prefer body-weight exercises now.
You were increasing the weights at a rate that your body couldn't handle. It is very frequent with new trainees because the neurological adaptations that appear at the beginning of training are always impressive: you finally learn how to use those muscles you always had and make them work towards a simple common goal, lifting that barbell. The main problem is that the rest of your body sometimes cannot keep up with that speed, and that results in this sort of issue. Had you been to a professional strength coach, he or she would have forced you to lower your pace and think about long-term progress over short term gain, which would probably have been at your 16 year old self's annoyance (as it was to mine).