Can you get sick with the same thing more than once? If I have a cold and then a few days later my boyfriend catches it, can he reinfect me once I'm better?

You . You can catch the cold again, however then odds are not very high. The reason being is that your body has made antibodies to the virus you likely had, and those antibodies may protect you when you are exposed to the same illness. It is a good idea to avoid close contact while dealing with an illness or right after, and maintain good hygiene; handwashing is key. Take care.
Theoretically, . Theoretically, no, due to immunity you've developed to that particular virus, but with your overall immunity down, you can become susceptible to other viral infections, or moreover, bacterial infections, and for example, can even end up with pneumonia, if you don't take care of yourself.