I place 3 containers in my backyard before a rain storm. All have vertical sides and completely open tops. I use a 1-lb coffee can, a 10-lb...

Rainfall is measured in terms of depth. That is why, we often say or hear (especially on the weather channel) that 8 mm rainfall took place or there are chances of up to 2 cm rainfall, etc. Rain falls, more or less, uniformly over an area (that is receiving rain from a particular rainfall event), that is, there is uniform distribution over a given area. Thus, the area of collection does not matter at all. As long as the collection surface is wide enough to receive the rainfall, has a uniform cross-section and is open, all the containers will receive the water to the same depth. 

One can also understand this situation by knowing that a container with larger cross-section will receive more raindrops, as compared to a container with smaller cross-section. Therefore size of the container does not matter.

Hope this helps.