I don't know what putnam is, but most of the time these mathematical competitions have nothing to do with research.
Being good in a mathematical competition means that you know a lot of theory (like the AM-GM inequality and others), that you have solved a lot of problems and that you have a quick mind.
However, research is (in my limited experience about something else). Of course you'll need to know a lot of theory to be able to do research. However, research doesn't mean that you can solve any problem in 6 hours. In fact, some problems will take you weeks to solve. It also requires a lot of creativity. Competitions want you to be creative too, but not so much.
I have met a lot of professors who claimed that they were really "slow" thinkers. That means that they had to do a lot of effort to understand a problem in all it's complications. That means that they would be very, very bad in competitions. Nonetheless, they were really smart and good in their research.
So, I would say: no, if you aren't good at competitions, then you can still be good in research.