Why doesn't each node store only a portion of the blockchain? 

For example, if you had 100 nodes, each node would store 10% of the blockchain. This would amount to significant overlap of each aspect of the blockchain, and as nodes came and left there would be increased or decreased sharing of the blockchain. 

Is it infeasible? If so, why? I am interested in pursuing this as a project because I think it could solve a lot of bitcoin's scalability/decentralization issues.

From my understanding of the blockchain, let's assume a particular node starts at the genesis block. Let's assume another node stores 1% of the blockchain past the genesis block. Couldn't multiple node verify block hashes for their shared 9% at a rate of once every 10 minutes? It wouldn't really use that much data -- you could hash the 9% of the blockchain to a particular value, produce a public key, and see if the other nodes can match a private key to said public key. 

A related question: Is this what Electrum does? Or, does Electrum simply store the entire blockchain on multiple servers that you can connect to?