The answer to your question is simply NO, unless it is really clogged to an extent you no more can ping the remote end (the best path peer). As long as you are receiving BGP updates from your peers, there is no way this can happen.
That was the simple answer. However, in some cases (depends on the configuration) your traffic may switch to the secondary peer. If your first peer (best path) link is continuously utilized to an extend that the remote end see it up and down, it may consider the link flapping, and if this peer (best path) has dampening enabled, then it will dampen your route for a period of time causing your router to switch the traffic to the secondary path.
We can go on and on for similar scenarios. I would suggest you manually distribute the load among both link by prepending some of your networks or changing the weight or something. Please check my reply to a similar question asked earlier in this forum,