Is it possible to catch the same cold twice. Well probably, but
unlikely in an otherwise healthy individual. Most colds are caused by
Rhinoviruses, which are members of the picornavirus family. While these
viruses are RNA viruses, their mutation rate it not very high.
Therefore, if a person gets over a cold caused by Rhinovirus A, and then
is re-exposed soon after, it is unlikely that the individual will become
re-infected. Immuity is strong against these viruses, and would be
expected to confer significant protection against re-infection, at least
over the short term.