Why won't my PowerPivot for Excel 2010 workbook upgrade to Excel 2013?

Recently we've seen an issue where some Excel 2010 workbooks containing PowerPivot models encounter errors when attempting to upgrade to Excel 2013. When opening a PowerPivot model that was created in Excel 2010 in Excel 2013 you will be prompted to upgrade with the following message:

This workbook has a PowerPivot data model created using a previous version of the PowerPivot add-in. You will need to upgrade this data model with PowerPivot for Excel 2013.