If it were possible for central banks to cut rates below 0%, would we still have experienced the Great Recession of today?
No, we would not have experienced the Great Recession. [except I cannot say that as I do not know]
However, there are lots of evidence pointing to the zero bound not being the real problem.
Other countries such as the eurozone and Sweden have experienced similar recessions, or worse, despite being above the zero bound for the vast majority of the past 5 years. You can write off the eurozone because of the cluelessness of the institution under Trichet, but Sweden is not so easy to write off. They did some very sound monetary policy in the first couple years of the recession, and then mysteriously gave up. And Sweden has a long tradition of great expertise in monetary policy and progressive policymaking.
The discussion in the US has not been what you would expect if economists actually thought the zero bound had caused the Great Recession. To begin with, the only reason the Federal Reserve has not cut IOR [Institute of Recruiters] further is that they are worried about the “break the buck” problem in the MMMF [Man-Made Mineral Fibres] industry. But if that trivial institutional quirk were widely seen as preventing the sort of faster GDP [Gross Domestic Product] growth required to avoid a Great Recession, there would have been massive demand for reform of the system.
Recall that the Fed got Congress to immediately give them authority for IOR in 2008 when they told Congress they needed it. As far as I know reform of the MMMFs was not even a part of Dodd-Frank. There is no evidence that mainstream economists, pundits, bloggers, etc, lose sleep over the break-the-buck problem, which prevents negative IOR.
So what is really going on here?
Lots of economists have not really thought through what they believe. At a certain level they believe the zero bound is the problem, but if pushed with solutions they fall back on other arguments. Those on the right would talk about “structural problems.” Those on the left would argue that highly negative rates would just blow up asset bubbles and lead to more income inequality. Very few economists actually believe the zero bound is the cause of the Great Recession, because very few believe that excessively tight money is the cause of the Great Recession.