Here's my experience with Visa/MasterCard offers:

If they say "Pre-Approved" (especially "Pre-Approved*") but have generic credit limit parameters such as "up to $50,000," then they will run a full credit check upon receipt of the application and you may be denied the account if there's something there that they don't like.

However, if they give you a specific credit limit (such as $3,000 or $4,500) that is clearly specific to your offer and not a boiler-plate number, then you'll almost certainly get approved if you return the application, unless something has changed substantially on your credit report since the offer was mailed.