After that, you have the "out-of-pocket" maximum. You pay 20% of costs until you hit the "out-of-pocket" maximum, which is typically thousands of dollars per year.
Beyond that, there are actually two different deductibles and two different out-of-pocket maximums. One for in-network services, and one for out-of-network services. You can go to an in-network facility and see out-of-network providers without notice.
And even beyond that, while it has been curbed with some recent legislation, if your insurance provider decides to pay less than the provider believes they are owed, the provider can bill you independently for the remainder.
So NO, it is definitely not $2k per year for a good health insurance plan in the US. FAR from it.