A year ago (and for 5 years running) Google bot probably did 2000-5000 page requests a day. I got hit by Penguin in Apr 2012 and lost about 50% of traffic but Googlebot continued to visit regularly. About 9 months ago I noticed Googlebot visits started to decline slightly. During this time my site was experiencing a slow lost of traffic. Googlebot also visited my site usually only on a couple of different IPs.
Since November traffic has stabilized (with some up and down flux) but Googlebot continued to visit less (300-1200 pages per day).
Have people found any correlation with rankings and how Googlebot crawls your site? Also in the last week or so Googlebot has started to visit my site from about 15-20 different IPs (verified as owned by Google). It hasn't visited any more pages it just uses more IPs to crawl my site.
There is a direct correlation between PageRank and Googlebot crawl frequency of a page. Googlebot will return and crawl higher PR pages much more frequently than low PR pages. As a page gains or loses PR, it will get crawled more or less frequently.
The correlation between crawling and rankings is less direct. Losing PR can cause both your crawl frequency and your rankings to drop. There are many other reasons that rankings may suffer other than loss of PR these days.