We generally do, really sleep during those nap breaks (and for what it's worth, we NEVER nap at home during the day, so it's a Disney-only behavior for us.) By the time we lie down around, say, 2:00pm, we've been up for 7-8 hours, walked several miles, and had a decent lunch. We turn off all the lights and draw the curtains so it's good and dark, strip down to undies and a T-shirt (or some other comfortable sleepwear), and turn on the air-conditioning so the room is cool, and we have some nice "white noise." In these conditions, the kids and DH are out like a light. There are some afternoons when I don't drift off for awhile (and sometimes, probably not at all), but I know that just resting quietly is better than nothing. We don't set an alarm, but we find that we wake up after an hour or two. We freshen ourselves up for the afternoon (put on a clean shirt if the morning was muggy, brush hair and teeth, touch up makeup, etc.), slip into our sandals (we tour faster and harder in the morning, so we typically wear sneakers then, and change into sandals for the evening - changing footwear mid-day also helps prevent blisters), and head out.
If we leave around, say, 4:00pm, we'll head to a park, do a couple of "anytime" attractions while awaiting our dinner reservation (generally about 6:30pm), have dinner (which typically takes less than an hour), and tour until about 9:00pm or so. That has us arriving back at the resort and getting into bed between about 10:00pm, having been "up and at it" for about another 6-7 hours. With full bellies and a few more miles' worth of walking under our belts, getting to sleep has never been a problem. We get up around 7:00am (the same time the kids would be getting up for school if we were at home -- it's great to keep them on the same schedule, at least for waking) and have a leisurely hour to "wake up slow," eat cereal in front of the TV in our PJs, get showered/dressed and head out for the bus stop. That means we're getting about 8-9 hours of sleep per night, which is actually an hour or two less than the kids and I (I need a lot of sleep; don't know why, I just always have) generally get at home, and another reason why the afternoon nap is a good and welcome respite.
I know everybody is different, and that for some folks, changing anything about their sleep patterns is difficult and more stressful than restful. This is usually the case with us at home, but we find that at WDW, when we're all getting plenty of exercise and fresh air, sleep comes very easily.