It must have been a free-for-all among collectors on Long Island, New York in 1880, by which time the Capped Bust half dollars (minted 1807-1836) had long since disappeared from circulation. In fact, such pieces had not been generally seen since the early 1850s, when many if not most remaining pieces were melted down for bullion when the price of silver rose on international markets.
The story is told by this item in the Evening Transcript, August 20, 1880: "A gentleman of Southampton, Long Island, writes that a great deal of curiosity has been excited by the sudden appearance in circulation of a large number of silver half dollars, all bearing the date of 1836, and as bright as when they came from the Mint. The mystery is thus explained: An old resident of Sag Harbor, formerly well known as a practicing physician, but who for several years has led a comparatively secluded life, at the time of the panic of 1836 [sic] hoarded 1,500 half dollars of that date. He kept them in total disregard of interest or premium until the present time. He has now put this hoarded treasure into circulation."
Actually it was the Panic of 1837, not the Panic of 1836, but in early 1837 (when financial problems had increased to the point at which most leading eastern banks stopped paying out coins) anyone seeking a large quantity of silver coins might likely have found them to be dated 1836. At the time the half dollar was the largest currently minted silver coin of the country, the silver dollar having been last minted in large numbers in calendar year 1804 (and bearing dates of 1803 and earlier). About 1,600 1836-dated Gobrecht silver dollars were made, but these were primarily placed into circulation in Pennsylvania and would not have been available in quantity to anyone seeking a bag or any other notable quantity of them at a bank in distant Long Island.
Today no 1836-dated half dollars are known with specific pedigrees to the Sag Harbor physician, but probably some of them did reach numismatic channels at the time of their release. The 2021 Guide Book lists the value of the 1836 Capped Bust with lettered edge from $325 to $925 in AU-50 grade and $2,000 to $5,000 in MS-63 grade, while the 1836 Capped Bust with reeded edge half dollar in MS-63 grade is listed at $20,000 and $5,000 in AU-50 grade.