We show how calculable IR loop effects impact positivity bounds in effective field theories with causal and unitary UV completions. We identify infrared singularities that appear in dispersion relations at |t| less than or similar to m(2). In the massless limit, they weaken two-sided bounds based on crossing symmetry, such as the lower bound on the amplitude for Galileon scattering. For amplitudes that are analytic in s even for large negative t, i.e., |t| >> m(2), we propose a new simple analytic approach to dispersive bounds, which are instead insensitive to the singularities, and explicitly compute the finite contributions from loops. Finally, we show that the singularity do not affect the bounds based on smearing in the impact parameter.