Two different round trip tickets work?

For various reasons, I would need to get this as two separate tickets.

I would have a round trip ticket with airline 1 between USA and HKN.

I then plan on getting a round trip ticket on airline 2 between HKN and Shanghai.

It would look like:

Day 1: Depart USA to HKN on airline 1
Day 2: Arrive HKN at 5am, Depart HKN at 9am to Shanghai on airline 2
Day 2: Arrive Shanghai
Day 3: Day in a Shanghai
Day 4: Depart Shanghai to HKN on airline 2
Day 4: Arrive HKN
Day 5 - 10: Time in HKN
Day 11: Depart HKN to USA on airline 1

Thanks!