Potencialmente, sim. Eles também podem não olhar para seus documentos. Mas é um risco.
Observe que (fonte):
If you are traveling to the US on a J1 or J2 visa, you will be
required to carry medical health insurance for the full duration of
your program. Many sponsor programs provide both their students and
dependents with health insurance, or you will be required to enroll in
a private health insurance plan. Either way, the US Department of
State has minimum J1 health insurance requirements that your insurance
plan must meet (which applies to J2 visa holders, as well).
Medical Benefits of at least $100,000 per accident or illness
Repatriation of Remains in the amount of $25,000 Expenses associated
with the medical evacuation of the exchange visitor to his or her home
country in the amount of $50,000 A deductible not to exceed $500 per
accident or illness A policy underwritten by an insurance carrier
with: an A.M. Best rating of ‘‘A-’’ or above; a McGraw Hill
Financial/Standard & Poor’s Claims paying Ability rating of ‘‘A-’’ or
above; a Weiss Research, Inc. rating of ‘‘B+’’ or above; a Fitch
Ratings, Inc. rating of ‘‘A-’’ or above; a Moody’s Investor Services
rating of ‘‘A3’’ or above;