I'm working on the upgrade project from 5.6.1 to 6.2. Initially I had the issue of Too many Sling api calls exception and my components are not getting loaded in the edit mode. Only when I increase the value of sling.max.calls to higher value, my components gets loaded.
I believe this is in the release notes somewhere (if I remember correctly). When I participated in the beta I noticed this issue and worked with product support to determine the same setup. In 6.2 the number of requests was lowered from the higher values 6.0 and 6.1 had by default. It should occur on a default installation if you keep the demo sites in place and add a decent sized custom site to your instance.