In the case of over-utilization due to a database query, moving to a server can either be a "band-aid" or further detrimental to the site. Legacy hosting has separate "web" and "database" servers, whereas a VPS would have both
on the same server. If this, along with the bad query, does not negatively impact the site immediately, it will as the site gains visitors.
In the case of over-utilization due to CPU or Memory usage, moving to a server can also be a "band-aid". The underlying issue with the scripting needs to be addressed prior to any upgrade. Left unchanged, the site will be negatively impacted as traffic increases to the site - the "when" will vary depending on the traDc and the server purchased.