If a Li-Ion cell has 3000mAh capacity shouldn't I be able to pull 3A from it for 1 hour?
In theory yes.
However, batteries have internal impedance and testing is a fickle thing. What can happen is that manufacturers use a test that actually runs the battery for for 10 hours at 1/10th that current and then uses those numbers to back-calculate what the 1 hour amp-hour rating. Since the 10 hour run operates at lower current, the losses in the resistor are less.