During last week's heavy rains in NYC, my below grade apartment (and
all others in my building) flooded due to backflow.  From what my
contractor tells me, it was due to the fact that the building's
drainage system is connected to the city's sewer and streetwater
runnoff systems, and it was too much for it to handle, so it created a
backflow problem that came out of all below-grade pipes in the
buildings.  My question is does this seem possible, and what can be
done to prevent it?  Specifically, are there valves that can be
installed between my shower, toilet, sinks and the building's system
so that this does not happen again?  What will a project like this
cost?

Valves could be installed at considerable cost, even automatic valves,
but a better solution is sewage pumps for all the below grade floors. 
 Neil

I agree with neilzero that valves could be installed.  They are
called, simply enough, "backflow prevention valves".  Depening on your
situation (concrete floors, most likely) the installation would be
very time consuming and expensive, especially in NYC, and might not
completely fix the problem (i.e. if there are leaky drain pipes under
the floor, the back flowing sewer could still seep up through cracks).
 Sorry to be the bearer of bad news...