I had to move from a Macbook Pro to a Macbook Air, and so I backed up my Pro using time machine onto an external hard drive. The backup ran smoothly and no errors were displayed. When I went to see the files in the "application support" directory (on the external hard drive using the new Air laptop), some folders "Steam" and "Minecraft" were missing. Is it possible they did not backup? Could they be hidden on the hard drive? I know the hard drive was not full. Also, I do not have access to the Macbook Pro.
Could you (or someone else) have setup exception rules for those folder so that they are not included in the Time Machine backup? (Possibly because they are too large)
You can either enable Finder to show hidden folder (just Google Finder show hidden folders) or view the folder from Terminal (ls -a).
Also, you may want to check the different "snapshots" of the Time Machine backup if it had been backed up before.