well it is not essential, but it can't hurt. I myself have never solved a Putnam problem, and did not aprticipate in the contest in college, but really bright, quick people who do well on them may also be outstanding mathematicians.
My feeling from reading a few of them is they do not much resemble real research problems, since they can presumably be done in a few hours as opposed to a few months or years.
E.g. the famous fermat problem was solevd in several stages. first peoiple tried a lot of sepcial cases, i.e. special values of the exponent. None of these methods ever yielded enough insight to even prove it in an infinite number of cases.
Then finally Gerhartd Frey thought of loinking the problem with elliptic curves, by asking what kind of elliptic curve would arise from the usual equation y^2 = (xa)(xb)(xc) if a,b,c, were constructed ina simple way from three solutions to fermat's problem.
he conjectured that the elliptic curve could not be "modular". this was indeed proved by ribet I believe, and then finally andrew wiles felt there was enough guidance and motivation there to be worth a long hard attempt on the problem via the question of modularity.
Then he succeeded finally, after a famous well publicized error, and some corrective help from a student, at solving the requisite modularity problem.
He had to invent and upgrade lots of new techniques for the task and it took him over 7 years.
I am guessing a Putnam problem is a complicated question that may through sufficient cleverness be solved by also linking it with some simpler insight, but seldom requires any huge amount of theory.
However any ropqactice at all in thinking up ways to simplify problems, apply old ideas to new situations, etc, or just compute hard quantities, is useful. I would do a few and see if they become fun. If not I would not punish myself.
