This answer is invalid for most people as of November 2013. The old rule might still apply to countries that have a visa waiver agreement with the EU, namely Antigua and Barbuda, The Bahamas, Barbados, Brazil, Saint Kitts and Nevis, Mauritius, and Seychelles.
I got the visa today, so I am answering my own question for the benefit of others.
There indeed is a 90/180 day rule for Schengen visas. But the way the stay duration is calculated is a little different from what is seen in most forums on the web.
The 6 month (or 180 days to be precise) period starts on the day of
the first entry into the Schengen zone (Note that the day of first
entry means the day you physically arrive in the zone and not the day
the validity of the visa starts). In that 6 month period, you can only
stay in the Schengen zone for a maximum of 90 days, irrespective of
whether you have a new Schengen visa issued by the same or a different
Schengen country that is valid beyond this 6 month period. At the end
of this 6 month period, a NEW 6 month period starts and you can again
spend a maximum of 90 days in the Schengen zone, provided you have a
valid visa. If your stay duration overlaps two 6 month periods, then
you must individually satisfy the 90 day limit in BOTH periods. All
following 6 months period will be calculated back to back from the
date of the first entry, until you remain outside the Schengen zone
for at least 6 months. When you stay outside for at least 6 months
(continuously) and THEN enter the Schengen zone, the six month period
again starts from the day of the entry. It would be as if you were
entering the Schengen zone for the first time.
All this is based on page 63 of the Handbook for the processing of visa applications and the modification of issued visas, issued by the European Commission, and my experience with this visa.