Well, the Patriots have drafted 14 tight ends under Bill Belichick. Trends have emerged. Size, obviously, is important. The "prototype" falls in the 6-foot-4 range and a shade over 250 pounds. Big hands (about 10 inches) are also preferred.
Athleticism matters, too. Quick 40 times (4.7-second range) and three-cone drills (7.0-second range) could help a player find his way onto the Patriots roster, as will legitimate lower-body explosiveness (35-inch vertical or thereabouts).
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Most importantly, they want hybrid players who can function as both receivers and blockers.