Rush Limbaugh loved to tell the story of how Jamestown died and then the Pilgrims almost died because of such a communist pact.
I'm not a Limbaugh listener, so I don't know what details he was providing, but this summary is incorrect.
The early English settlers were not communist, and did not enter into a pact preventing private property. Rather, the overwhelming majority of them had signed on as
indentured labor, meaning that they were working for someone else's profit. Unsurprisingly, since there were strict rules against mistreating indentured servants and the terms of their contract prevented them from having personal money or private goods, that meant the local economy was incredibly stagnant.
It is unlikely that this is why Jamestown died, but it certainly was a contributing factor to the difficulty many early colonies had with establishing independent markets. When contracts were revised to allow indentured servants to possess personal property, local businesses (as you might expect) boomed.