The Fed could certainly have limited the bubble by raising rates sooner coming out of the 2001 recession. Unfortunately by 2005, the bubble was mostly inflated - a stronger response in 2005 or 2006 probably would have just moved the 2008 crash up by a year.

The Fed's regulatory powers before Dodd-Frank were limited to large banks. S&L's had a different regulator (Office of Thrift Supervision) which was more lax than the Fed, broker / dealers were regulated by the SEC, and the subprime shops were regulated by no one, basically. The OTS went away after most of the large thrifts that it was supervising failed (those thrifts not only originated some of the highest risk products like option ARMs, they then held onto them rather than selling them like the investment banks did.)

After Dodd-Frank, the Fed has much more control - most of the brokers are now banks, and the Fed can now regulate any large company that it deems to pose a risk to the financial system.

Lately, the Fed has ostensibly been trying to limit the amount of risk that the banks can take on by raising the amount of capital that they're required to hold, both generally and specifically for whatever types of financial assets that the Fed considers "high risk", such as construction loans. The banks have actually cut back on some of their trading businesses for which the new rules are especially punitive (emerging market debt, some of the more arcane securitization businesses, etc.) because they can no longer earn enough of a return on the higher amount of equity that they have to hold for those businesses.

Of course, if the monetary arm of the Fed is doing whatever it can to reflate asset prices, we'll see how well the regulatory arm can do in controlling the potential damage...