I think this question is a very subjective one and the answer can differ from person to person. Usually people associate generosity with money but I think that generosity comes in many forms. One person can be financially generous to others while the other person can be emotionally generous.
Then there is also the fact that a person can have many aspects of generosity while another person might just have one aspect of generosity. Nevertheless, I think that with concentrated effort, a person can acquire one or more of these aspects of generosity.
Like all worthwhile things, acquiring the trait of generosity may not be an easy task for somebody who may not be generous by nature to begin with. Important thing to keep in mind is that it takes practice, but above all you would need a good enough reason to start practicing being generous. One can start slowly by sharing things that they might find difficult to part with. This would only be a start and the next step would be prepare yourself mentally and decided the limits of the generosity you want to acquire but without hurting yourself too much.
Personally I feel that staying in the company of generous people can help one acquire the trait of generosity.