T cells and B cells are able to recognize all kinds of antigens (i.e. stuff displayed on a viral or bacterial surface.) These immune cells can recognize different parts of the same virus, so there will be multiple cells that can recognize the same virus. T cells and B cells in general may not even see the antigen to which they are targeted within their life span. These cells have a limited life span and will die off. Not to worry though because your body is constantly making these cells every single day. Only memory B cells have a long life span and I'm not sure that it is certain exactly how long they can live. Memory B cells are B cells that have recognized their antigen (for example, a viral surface protein) and have also been stimulated by a helper T cell. These memory B cells make up your adaptive immune response and allow your body to respond quicker to the virus should you get infected with it again. So, for example, the flu vaccine will help stimulate the creation of memory B cells. If the real flu virus tries to infect you, you can make antibodies against the virus quicker than if you had never been exposed to it before (hadn't gotten the flu vaccine.)Thus, you won't get as sick or may not even get sick.

The below information was obtained from the web and explains the main differences between innate and adaptive immune responses involving the different types of lymphocytes (white blood cells), to which your question relates. B and T cells are only developed against a pathogen to which the body has been previously exposed. Essentially if you havent been exposed to a particular pathogen before, you do not possess specific populations of B and T cells against it. It is within this statement that I think your possible confusion arises. The word that is missing from your original question is "type". You do not simply possess a single T or B cell against a particular pathogen, but rather one type of T or B cell (meaning that you possess multiple clones of the same cell type).

The three major types of lymphocytes are the natural killer (NK) cells, T cells and B cells. NK cells are a part of cell-mediated immunity and act during the innate immune response. They can attack host cells that display a foreign (e.g. viral) peptide on particular cell surface proteins known as MHC class I molecules. Once they determine a cell is infected, the NK cells release cell killing (cytotoxic) granules that will destroy the infected cell. NK cells do not require prior activation in order to perform their cytotoxic effect upon target cells. Like NK cells, the T cells are chiefly responsible for cell-mediated immunity whereas B cells are primarily responsible for humoral immunity (relating to antibodies). T cells are named such because these lymphocytes mature in the thymus; B cells (named for the bursa of Fabricius in which they mature in bird species) are thought to mature in the bone marrow in humans. T and B lymphocytes differ from NK cells in that they are the principal cells involved in the adaptive immune system. These are cell types that retain a memory of a previous infection so that they can respond to the same infectious agent quickly upon reinfection. In the presence of an antigen, B cells can become much more metabolically active and differentiate into plasma cells, which secrete large quantities of antibodies. T cells, after they see an antigen, will also become highly activated and will secrete specific proteins, such as cytokines and cytotoxic granules, depending on their subtype/function.