When I started going to gym, it happened that I had chest+shoulders+biceps day, back+triceps day and legs day. But I've heard many times that people tend to do chest+triceps because triceps works during bench press and the same for back+biceps during pull-ups. I tried doing triceps after chest and were able to do fewer reps(on dips) and less weight/reps on crossover triceps. So, is it really good to combine chest with triceps, even if weights decrease?
Well, I would ask you this simple question : What is the point of your training ?
Is it doing as many reps as possible, or is it building more muscle ?
I would guess the second one is your concern. In that regard, what you are looking for is destroying you muscle fibers in order for your body to overcompensate for your losses.
Remember this :
When it comes to building muscle, weight is a mean to the end, not the end in itself.
So obvioulsy, you will be able to do less reps, and that is a good thing. Most bodybuilders don't care about the weight they are pushing, as long as they feel it stresses their muscle enough.
When you are really tired at the end of your workout, it is normal - and a good thing - to be able to push way less. A good example would be doing drops Sets.
That is why a split makes a lot of sense.
What I would tell you is that for upper body, chest, shoulders, and triceps usually work together(bench press, dips, push ups, shoulder press, chest press, tricep dumbbell bench press(see bodybuilding.com)...) well back and biceps usually also work together(pull ups, lat pull downs, curls, rows, chin ups, iso row, etc.) Legs and abs are a different category, but to answer your question, yes, chest and triceps should be worked on the same days.
Yes, chest and triceps have to be worked on the same day, due to the fact that nearly every chest exercise works the triceps, too. The same law applies for the back and biceps. Imagine a robber walking into a bank without a gun. There is no robbery, because there is a missing link, the gun. Trying to rob a bank with no weapon is like trying to work your chest without your triceps. You need to do both to have a successful workout.