Say for example, I have a 5 V 1.0 A Source, and 2 x 3300 uF capacitors in parallel, and the output is drawing 5 V, 3.0 A, can it accommodate the 3.0 A requirement of the output?
Yes, for a very short time but otherwise no.
A capacitor in this configuration behaves a bit like the water tank in your toilet. It stores charge and can deliver a short high volume flush but has to be recharged at the rate the supply can manage and this takes some time. Flushing again to soon doesn't work as the tank hasn't refilled. Adding more capacity to your tank (parallel capacitors) increases the volume you can discharge in one flush but the refill time increases proportionally. In your case the maximum fill rate is determined by the current limit in your PSU.
Figure 1. A smoothing capacitor smooths out the pulses from a rectified AC supply. Source: WikiMedia.
Usually we use capacitors in the configuration you've shown to smooth out a rectified AC supply as shown in Figure 1. The graph shows that the capacitor gets charged on each half-cycle and keeps the voltage up during the dips between pulses.
For more on this see Electronics Tutorials Full-wave rectifier