This is probably a better question for /r/AskScience, but the answer to your question is generally no. The bacteria Yersinia pestis causes bubonic plague. If someone survives the somewhat crummy mortality rating, they will now have particular T-cells in their bone marrow and their entire purpose is to destroy that particular antigen when exposed to it again. That is how it is supposed to work, but if someone has an immunodeficiency or some other health issue the bacteria can still overwhelm the body again and they would have bubonic plague again.

With the numbers of people who died from it, statistically someone probably did get it twice, but a few out of fifty million is pretty negligible. With that said, someone may have more insight into reports of what you described as happening.