Huh, I thought I had replied to this thread, but I guess it didn't go through. Anyway, yes, heavy compound lifts will often affect the exercises that come after them. Obviously, exercises that overlap a lot will interfere with each other quite a bit. In your case, the bench press and dips both rely primarily on the chest, triceps and shoulders, so doing one will obviously impact the other. But even unrelated exercises, such as the squat and bench press, may see a small impact for the following reasons:
1. Heavy compound lifts use MANY muscles through out the body, so some overlap is often inevitable. For example, the squat hits the quads, hamstrings, glutes, abs and the entire back. The deadlift uses almost the exact same muscles, so the overlap there is obvious. However, even during bench press, the muscles of the back and abs come into play to stabilize the weight.
2. In fact, when doing heavy compound lifts, you are likely to tense ALL the muscles in the body. This is especially true during attempts close to your 1RM where every fiber in your body will be straining.
3. Weight lifting can fatigue the central nervous system (CNS). Heavy squats in particular can take a heavy toll on your CNS and I wouldn't be surprised if this was the biggest reason they may have at least a small effect on every exercise that follows.
4. In general, more exercise means more fatigue. Hell, if you bike fast for an hour, you probably won't bench press as well, even though biking just uses your legs. I'm not sure at what level this fatigue is working - overall energy levels, glycogen stores, or maybe again CNS fatigue - but I'm sure we've all felt it.