You probably need to strengthen your upper back. Do some one-armed bent over dumbell rows, some cable rows, something like that. If you want to know
why, read on.
Often when you do one lift with one set of muscles doing the pushing or pulling, another set of muscles come in to stabilise everything for you. They hold things in place so your muscles can work best. Let's look at the triceps brachii, the muscle you're working with the pushdown.
Your triceps brachii have three heads - thus their name. The "long head" has its origin (anchor) on your scapula (shoulderblade), the "medial" and "short" heads on the top of your humerus (upper arm bone). They insert on your ulna (the lower arm bone that runs to the little finger side of your hand). Their action is thus to pull on the ulna, extending the elbow.
However, when they pull on the ulna, they also pull on the other end. This happens with every muscle - that's why if you do a leg curl, your bum rises in the air; the hamstrings pull on the lower leg, but pull on the hips, too.
Where the triceps brachii pull on the humerus there's usually no problem. But where they pull on the scapula, left to itself the scapula would jump forwards - so then there'd be no tension left in the triceps long head, and you couldn't extend your elbow - the weight wouldn't move.
So some other muscles have to hold the scapula back. That's why when you do a triceps pushdown your upper back muscles contract. They're holding the scapula back. There are a whole swag of muscles (too many to picture here) which retract your scapula (hold it back), many of them among the often-injured "rotator cuff" muscles.
If you do a lift which the agonist (working) muscle can handle, but the stabiliser (supporting) muscles cannot, the stabliser muscles may be strained, and hurt.
It may be that while your triceps brachii are strong enough to push down 60lbs, your rhomboids, supraspinatus, etc are not strong enough to hold the scapula back against 60lbs of force. This is very common, why is why the rotator cuffs are so often injured.
The solution is to work the stabiliser muscles to strengthen them. For example, it's a healthy (for posture) goal to be able to row as much as you can bench. If you can bench (say) 20kg dumbells, you should be able to row them, too, for the same number of reps.
It is very common for people's - especially men's - backs to be weaker than their chests. This is because of our modern lifestyle, lots of time at the desk and not much picking up heavy sht off the ground.
Work your back harder.
