Big fish are the ultimate target of many, if not most, fly fishers. Big fish aren't necessarily smarter though, they are often just overlooked.
A few simple things to remember when targeting big fish is to fish where big fish are and to fish when big fish feed. I know this sounds simple, and maybe even a little "no duh", but many anglers miss out on the biggest fish by fishing in the wrong spots and at the wrong times. This obviously varies quite a bit by species and location but with trout, being territorial feeders, most big fish occupy the best feeding lies.
In good hatches this may be the tailouts of runs, lies closest to the banks in slow water, or the best current seams. At other times it will often be at the heads of runs where they get the first shot at the food. With browns, cuts, and brookies the biggest fish often lie closest to the best cover or deepest water. With rainbows the biggest fish often occupy faster deep water where more food drifts by. If you're catching a lot of small fish it is likely that you aren't going to catch the biggest fish in that particular spot. Also, big cutthroats, brook trout, and especially browns are notorious night feeders. Staying late or arriving early or simply fishing all night can result in the biggest fish in the river or lake you like to fish.