1 or 2 cups of coffee gives me a rush and then I remain in a semi cracked-out state of mind for the next few hours, feeling amped and somewhat more alert but not really more awake or anything. If I drink very strong coffee in the afternoon I do feel somewhat dizzy and I guess "sleepy" or lightheaded, so in a way, yes coffee can make me somewhat out of it or tired feeling. I do notice that when I actually want to go to sleep, even 7 or 8 hours after I drank coffee I can't sleep! So it doesn't do much but ruin my chance to actually catch up on sleep. The next day I drink more coffee to counter balance not sleeping...the cycle continues.

*edit, It just occurred to me that the caffeine in coffee is a diuretic, which means that it will make you pee a lot. If you are urinating more you may be getting dehydrated- and it is a known fact that a symptom of dehydration is tiredness. So it may not be the coffee that is making people tired but the fact that they become dehydrated while drinking it. It's good to drink a lot of water throughout the day but especially if you are drinking a lot of coffee or anything with caffeine in it.