Of course, why couldn't you? It's an asset just like anything else. It isn't a security where wash-trading rules would apply to keep people from selling at a loss and immediately buying. But every time you buy or sell it (or almost any asset), there is a taxable event, where you owe taxes on the gains (and can deduct the losses). Nobody is likely to care if someone is playing with a small amount of money and doesn't self-report it. But if they are buying/selling material amounts of money's worth on a US-regulated exchange - and don't track their taxable events, they are in for a nasty surprise if/when the IRS gets curious about their activity.
Anyone who owns bitcoin through another entity other than just having the keys themselves, is subject to whatever rules that entity has in place - whether it is a mutual fund through their non-taxable retirement account, one of the newer ETF's, or any of the rules that the exchange itself has.