Lambert family of Panfield Hall

I have some Lambert ancestors who supposedly lived at Panfield Hall in Essex. I have come across another Lambert family who also lived at Panfield Hall (that of John Lambert, a Puritan who immigrated to the U.S.A. with his family). I'm guessing that these two Lambert families are related somehow (Panfield Hall wasn't that big, after all), but my searches have turned up nothing. John Lambert was probably the contemporary of my Jeffrey Lambert; but I don't know if they were brothers, first cousins, second cousins, or even more distantly related.

It's amazing how much information is online these days. But it is still possible to come away from an Internet search disappointed... especially when you're researching things that happened hundreds of years ago.