I would like to insert multiple instances of the same block, but I'd like each instance of the block to be independent of the others.  If I edit one of the blocks, I don't want it to affect the other instances of the block.

Hello CADnStl, I'm afraid that it will not be possible with one unique block as it's the aim of the block theory to be a "standard" part to be updated as one.A posibbility would be to do block(s) inside that block with a block to contain each and every common items and that particular block to be present inside other blocks with slightly different names.Block name including the common items : AAAOther blocks names with more items : AAA-1 , AAA-2 , AAA-3, etc...

It is nonsense. If you want to use instances of the same block, you will of course change all instances if you change the definition of that same block. Or are you talking about editing the references? (like location, scale, rotation, attributes)If you want to have references of different blocks, you can make copies of the original block definition.

If you edit a block definition, it will change all blocks in your drawing. Unless if you meant you can change the block appearance/size with Dynamic Block behavior.