No. Bitcoin's blockchain solves the double-spending problem with the proof-of-work. They key of this is that nodes only accept the longest blockchain, or the one with most work put into it, as valid.
If the blockchain were cut every 4 weeks or so, you would have many short chains. How would you reach consensus in the network on which chain to use? Miners could easily build chains longer than 4 weeks using the past chains, and overwrite the current one with new transactions. Enabling them to double-spend and completely invalidate transactions on other chains.
Essentially there would be no way to reach consensus on the chain to use. Currently the consensus is reached by using the longest chain.