I'm bigger than your boyfriend (370-380) and I would not ride a dept store bike if only for the fact that a used bike shop bike will have quite a bit better components than the dept store bike.

Most dept store bikes use freewheels in their rear wheels and fat guys like me have a tendancy to snap axles or bend axles due to the design of freewheels. On the other hand, even a lower end Trek or Specialized or Diamondback or other local bike shop brand will have wheels that use freehubs and not freewheels. I have been riding for year now and have not bent any axles or broken an axle.

The big weak link for big people are the wheels on cheaper bikes. Other than that, its just a matter of keeping your bike maintained and working well.