Your bench is like a chain. A chain is of course only as strong as its weakest link. If your weakest link is triceps, and they will only let you press 300, but your pecs, shoulders, lats, etc. will allow you to bench 400, guess what? You are a 300 bencher. You don't get to split the difference.
Once you have caught up your tricep strength to the point they can handle 400, then something else may rear its ugly head as the new weak link. Maybe it is shoulders. (My personal weak link) Now to get forward progress, you will have to strengthen the shoulders to the point they can handle as much as your triceps.
It keeps going, once you strengthen one muscle group, then another may become your new weakness. For a competitive bench presser, the equation is usually simpler than for a gym trainer. Most gym trainers don't use a shirt. Competitive guys do use a shirt.
For a shirt wearer, the lockout, or triceps is almost always the weak link. Modern shirts help a great deal out of the bottom of the press, where shoulders, and pecs tend to show their weakness (or in some of us, injury), but the shirt "boost" quickly runs out, and it is up to those pistons in the back of your arms that mortals call triceps to take over and lock the elbows out.
But the rule of strengthening the weakes link and continuing to do so applies to everyone. Also, don't forget that the person who moves the bar the fastest usually locks out the most weight.
Hope some of this helps, and good luck.
B.