The problem isn't limited to Psalm 62; a small sampling of other Psalms show the same numbering issue. The text file I used for MSG doesn't appear to have any formatting issues on either 62:1 or 62:2 (where the numbering goes wacky). Weird stuff.
As I stated earlier in this thread, MSG appears to follow the KJV numbering of Psalm verses. So, if you copy the entire KJV verse mapping for the psalms into the MSG verse mapping, MSG should display properly with other versions throughout the Psalms. I suspect that MSG follows KJV verse numbering throughout the Old Testament; that is why I suggested simply copying the entire KJV verse mapping into MSG's verse map file. But I do not have the Message personally to check out my hunch. So, yes, the misalignment is more than in just Psalm 62.