A total of 68 people associated with the Andover Subacute and Rehabilitation centres have recently died, with 26 having tested positive for Covid-19.
Police did not find a body in the shed, but said the facility's tiny morgue was "overwhelmed".
New Jersey has over 71,000 cases and 3,100 deaths due to the coronavirus.
Over the weekend, the nursing home had requested 25 body bags from authorities. On Monday, police received the anonymous tip about a body being kept in a shed.
Instead, they found 17 bodies kept in a morgue built to house four.
"They were just overwhelmed by the amount of people who were expiring," Andover police chief Eric Danielson told the New York Times.
It is unclear whether any of the 17 deaths were due to the coronavirus.
Chaim Scheinbaum, a co-owner of the nursing home, addressed the morgue problem in an email to New Jersey Congressman Josh Gottheimer, according to the Associated Press.
"The backup and after hours holiday weekend issues, plus more than average deaths, contributed to the presence of more deceased than normal in the facility holding room," he said.
Mr Scheinbaum also said the facility is adequately staffed.
How big is the problem in care homes?
Seventy-six patients have tested positive for Covid-19 along with 41 staff members between the two buildings, according to the Times .
Thirteen bodies were moved to a refrigerated truck at a neighbouring hospital, while the remaining four were to be sent to a funeral home.
The nursing home owner has since obtained a refrigerated truck for bodies, local media reported.