Can I get a the same bacterial infection more than once?

Yes. Some bacterial infections don't create a strong memory or immunity in our bodies. Strep throat infection is such an example. School children sometimes catch strep throat from classmates every one or two years for several years. Gonorrhea, an std, is another example. A person can catch gonorrhea, get antibiotic treatment to clear it, and then catch it again a short while later.
Yes. Unlike some viral infections. You can get repeated infections with the same strain of many bacteria. Even with bacteria which induce the development of protective antibodies like strep. Pneumoniae you can be infected by different strains of the same bacteria.