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The story of his noble birth can be found in many sources (not the least of which is the The Hastings Memorial). It is not clear where this story started but there is no evidence that Thomas ever made this claim for himself. The specific assertion that he descended from the Earl of Huntingdon branch of the Hastings family is not supported by any of the available evidence (and there is quite a bit on this well documented noble family). Looking at geography alone, it is highly unlikely that he would have come from central England (the Huntingdon branch was seated in and around Ashby-de-la-Zouche) when the vast majority of his fellow immigrants were from East Anglia. It is highly probable that he was also from East Anglia and probably of the artisan class. For more details read "The Supposed Noble Ancestry of Thomas Hastings."
The family origins of his first wife Susan have not been ascertained. When or if they are this may be the critical clue in establishing his own place of origin. The problem affecting 17th century (and earlier) researchers is that records on all but the most prominent families are sketchy at best. They weren't great (or highly legible) to start with and during subsequent centuries they have been lost, damaged or just subject to general decay.
Thomas did own property in Dedham during the mid and late 1630s but sold it when that town established a residency requirement. There is no evidence that he ever lived on his lands in Dedham.