I think this is an underrated question. It gets to the heart of training and by the range of answers you have had so far, I think it is obvious many disagree on the subject. I was exactly the same as you in high school. I had split 50-52 on 4x4 relays consistently. I mostly ran the 1600. I had difficulty closing out races. I have also coached many athletes with similar times. Here's what I did as a coach to get that last 400 to 600 more successful. I never really learned how to do it in high school due to improper coaching.
Training:
I don't think there is one answer to fix your issue. A strong aerobic system is critical to the race. The 1600, although a short fast race, is still mostly dependent on the aerobic system. If you have been doing decent long runs (at 7:00 pace or a little slower) consistently for weeks it can help. As a coach, we kept our longs runs all the way to the end of the season. For the 1600, though, a great deal of stamina is required (the ability to run a hard pace and hold it). Early season we had a consistent diet of threshold work. For you that would probably be in the 5:20-5:45 pace range. Then as we moved into the season we did a lot of pace work at 3k to 5k race pace. We tended to really push the volume on this (5-8k worth, which may have been too much), but my runners were rock solid in the middle portion/late portion of the race.
Mental:
You really do have to practice visualizing yourself closing races the way you want. I would suggest cut-down workouts in which you increase your pace over the final part of the workout. 200s, 300s and 400s at 1600 race pace are good workouts to practice this. But you visualize this anytime you are just thinking about your races too and I think you change the way you race. The mental is just as, if not more, important than the physical.
I think if you can do this, you will unleash your raw speed and be running under 4:25. Possibly way under that.