Hi,
You should take a close look at the Release Notes of each version. They give direction on what you're asking for as far as whether you can upgrade directly from 6.2 to 6.3.2 as well as any changes in configuration that you should be aware of. For example, when upgrading to 6.3, obviously you get the added feature of Multi-Team Environments, in which there are new permissions involving web services and folder/component level permissions. I'd look over those release notes carefully before upgrading.
However, in general as I've seen in the release notes, you can usually jump straight from one version to another if it is within the same major version (i.e.- 5.1 to 5.3 OR 6.1 to 6.3). I'd still verify the release notes to be sure though.
Dave Morris, 3Ci at Southern Company