Drives don't all fail at exactly the MTBF time: rather, the times at which they fail obey a particular statistical distribution with the given mean. You don't necessarily need to test for as long as the mean to get bounds on the mean, since testing for a shorter time can still give you a lot of information about the shape of the distribution.
For example, suppose you want to demonstrate that the MTBF is greater than one month. If the MTBF was only a month, you'd expect a few drives to fail very quickly so if you tested a bunch of drives for a week and none of them failed in that time, you have reasonable grounds for believing that the MTBF is quite a lot more than one week. If you test enough drives for time T, you can argue that the MTBF must be at least some larger value.
Also, they may be using an argument along the lines of "We tested the drive by reading and writing 24/7 for a month. In reality, most users only access the drive for 1% of the time that the computer is running, so most users will experience one hundred times the MTBF we found in our tests."
Another technique that may be used is to test in harsher conditions than real use. I don't know if this is used for hardware but it is used for shelf-life of foods. First, you do experiments that show, for example, that your canned whatevers degrade three times as fast when stored at 40C as they do at 20C. Then, if they're still good to eat after four months in storage at 40C, they should be good to eat after a year at 20C.
Normally it's just a number for marketing, not something that has been certified independently and they don't have to back it up with anything. Plus its quite possible there could be a firmware change or something that causes thousands of units to fail within a few hours. But depending on the manufacturer, SSD's sometimes have a maximum total bytes written value specified (e.g. 75TB) and this is more relevant because exceeding it can effectively void the warranty. Typically though it would be very difficult to write that much data in the warranty period. – James P – 2015-08-05T11:58:12.833