Is there overbooking and voluntary deboarding in First Class?

In coach, I sometimes try to get a reservation at a busy time, and volunteer in case they have overbooked. But if they haven't overbooked, I end up flying in jam-packed coach, barely able to breathe. So other times I try to fly at some uncrowded time, though there is no chance for a free ticket then.

I've recently started to get serious about FF miles, and realized that by changing phone companies, stockbrokers etc. I may get so many miles I can afford First Class, especially when that is as little as 40,000 miles. Since First gets no more crowded than having one seatmate next to your wide seat, which is quite OK, maybe I should book for a crowded time and either get a free ticket or at least fly without discomfort.

But someone told me they don't overbook First, so there is no chance of a free ticket. Is this true?