You should be able to before you ever think about doing more direct arm work. There's no better relative strength test than the strict pull-up, and there's no reason to do direct arm work if you can't.
A strict pull-up starts at a dead hang position at the bottom of the range of motion, and ends with the chin moving over the pull-up bar. (The neck remains in a neutral position; don't "reach" with the chin.) Any momentum from the legs or torso makes it NOT a strict pull-up.
Take this test with objectivity and see how you stack up. If you can do eight or more strict with any hand position other than a mixed grip (overhand, neutral grip, underhand), you've earned the right to start doing direct biceps and triceps work. Your base of relative strength is at a level where you'll actually benefit from these more isolated direct arm movements.
If you have to cheat to do all eight reps, you can either continue hitting biceps and triceps, gravitating towards inherently easy lifts, or you can set your ego aside and work on getting strong in the compound lifts. Choose the latter and you'll be surprised by the type of full body strength you'll be able to develop. Chances are, in the process, you'll also build bigger biceps, triceps, and forearms than you ever did doing dumbbell curls.