Then this needs to be your primary concern.
Fix your posture.
Execute exercises with proper form and range of motion
Strengthen your core and learn proper bracing
If you do not do those things trying to add weight to the bar and beating the logbook won't always be your best course of action. Fix the underlying issue, and then when you go to progress in the gym everything will be firing on all cylinders. This way you can avoid injuries while trying to add strength and add weight to the bar.
Spend a session or two with a coach before you cement bad training habits. As you get stronger you'll risk severe injury if your form is not on point.
You can train for an emphasis on strength and look great, it all comes down to body fat. You'll always be stronger at a higher bodyfat % than you will shredded but that's no big trade off if you're not competing in power lifting. 5X5 is a great program and will take the average lifter a long way. There are plenty of strategies one can employ to break a plateau. Google 5x5 plateau and take your pick when you get to that point but I'd imagine you'll be able to build up to a good base 3/4/5 bench/squat/deadlift before you need to tweak anything.
I am going to second/third these guys. Some of what I'm about to say may take some flack, but you don't hurt yourself lifting weights. You hurt yourself throwing and jerking weights, twisting, contorting, etc. If a weight is too heavy, you simply cannot lift it. If you cannot lift it, and you try to use another method to move it, you risk injury. For instance, pack on 1000 pounds onto a barbell on the floor. Now try to hurt yourself dead lifting it. (Ok, don't ACTUALLY try this - it's a thought experiment).
Focus on your issues first, get things right. Don't focus on how much weight you lift as much as HOW you lift it. Leave your ego at the door.
Now, to your original question, I originally thought, "Of course not. What is this foolishness?"
Then, it occurred to me...."Yeah, OK, maybe you could be too strong."
For instance, I believe I saw that Eddie Hall has a myostatin deficiency and he just broke a deadlifting record of something like 1100 or 1200 pounds. When he did this, he ruptured blood vessels and had nose bleeds, etc. Clearly he has built enough strength his body isn't really capable of handling the stress he can generate. Someone who could possibly go further may snap bones from their strength.
But is this anything that 99.9999% of us should worry about? Not even close. I mean, if you can deadlift 50-70% of that weight you will be in the top 1% by a long shot, and competitive with all but the most elite lifters. And do you have a "disease" that helps you pack on muscle, plus the dedication it took for him to get there (and the "supplements").
So, yeah, anything is possible. Is it likely? Exceedingly unlikely.