With the exception of one time policies like travel insurance or short term policies most health insurance policies don't have maximum annual benefits. The maximum is expressed as the lifetime maximum, usually $1,000,000 to $5,000,000 depending upon the company and plan.

For the sake of argument even if you could purchase policies as you describe I don't see where splitting the risk into 3 levels reduces the total cost. The 3 policies would probably exceed the cost of a single policy because you have the expenses of administration, underwriting and policy issuance times 3 and that's before any of the companies factor in potential losses.