It is possible, provided that you get a visa as you describe. However, you might not want to book separate tickets like this.
There are a few ways to book this trip, and while all will "work," some may be substantially more expensive than others:
- One multi-stop ticket (single purchase transaction) covering (SEA->BKK; BKK->DEL; DEL->SEA).
- The same, except three separate tickets. SEA->BKK; BKK->DEL; DEL->SEA (as there are no direct flights from Seattle to Bangkok, you'll need to stop somewhere else on the way)
- Two separate tickets. SEA->BKK and (BKK->DEL; DEL->SEA)
- Two separate tickets, as you propose. (SEA->BKK; BKK->SEA) and (BKK->DEL; DEL->BKK)
The problem with booking separate tickets is that you, not the airline, are responsible for your own connection. If your flight is delayed, you could be responsible for the cost of rebooking, while with a single ticket, the airline is responsible for getting you there even if you miss your connection. You'll usually have to claim and recheck baggage, which can require passing through immigration and customs (which may require a multiple-entry visa). If baggage fees apply, you'll have to pay them again. Also, if you need to change your flights, you'll likely need to pay multiple change fees, as each ticket is separate.
And in this case, since there are no direct flights from Seattle to Bangkok, it's not clear that it makes logical sense to fly back through Bangkok for your return flight; there are many one-stop routings between Delhi and Seattle.
Particularly if you have checked baggage, I'd avoid #4, because you'll likely need to claim and recheck your bags during your return flight. If you do book separate tickets, I'd allow a long time between connecting flights to allow some margin for delays, and obtain a multiple-entry visa so you can enter the country on your return trip as well.
Finding good fares on multi-stop airline tickets is tricky (and sometimes impossible), and requires searching a lot of different combinations. You might consider working with a travel agent (or a service like Flightfox).