I'm not sure if this is intentional but for at least 1-2 seconds after the reload animation finishes, my sniper refuses to shoot. This has lead to many frustrating moments in the crucible when my gun doesn't shoot. This has been a problem on both PS4 and PC, and I haven't seen anything else on this subject so if it has been addressed before I apologise.
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Wait for you character to actually finish the reload animation I remember noticing this awhile ago. If you just wait for the clip to go in and your ammo count go up, and then run to cut the animation short, it messes it up. You have to wait for it to actually pull the handle back and finish reloading, then you should be able to fire right away