I would speculate yes...
One of the benefits of BCAA supplementation before working out is that it competes with the same receptors that process serotonin, one of the main neurotransmitters that creates the onset of fatigue. Too much supplementation throughout the day could lead to low levels of serotonin which are also responsible for processes of mood control (happiness) and sleep. This is just one thing I could forsee being a problem, but I'm sure there is more.
Pro-tip for BCAA supplementation: I would argue the best utilization of BCAA's is not pre/post-workout or before a meal but in between meals. You can have high blood protein levels but your body will shut off mTOR after a certain period, BCAA's (and particularly Leucine) are responsible for mTOR activation. Hence supplementation a couple hours after a meal will kick-start muscle building, theoretically allowing you to have constant mTOR activation throughout the day.