Without sounding stupid – it depends on the changes.
The biggest issue when upgrading is not just the ‘look and feel’ of the theme, but all the functions that are used by the theme. For example if your theme uses add_option() it may not work, later versions use update_option() instead (as it creates the option if it does not already exist).
As we do WP work I personally always download a new WP version onto my local PC, install it and run my themes on it using the WP test data before updating, probably a luxury for most users, but it is worth the extra time for us.
Failing that ensure you have saved a complete backup of all files and the database before upgrading, as I would say there would be a fair chance of a bespoke theme failing on upgrade.