Here is a bit of detail on the exchange, as I've emailed the Gillette ticketing office many times to clarify (and I must say, they have great customer service with prompt and detailed responses if you ever have questions).
STH and Waitlist members can sell there tickets on the exchange, and can purchase up to 4 tickets to any game. It is the only approved method to sell your tickets from the Patriots, though they have no issue with you selling or giving tickets to friends at face value (they explicitly told me this), provided the attendee obeys the rules, as they are representing the STH. People that are overly intoxicated, fighting, etc. and are removed from the game can cause the STH to lose their seat. They did tell me there is generally a warning, and it very rarely occurs.
The Patriots also put tickets on the exchange from time to time that are freed up from other sources, these can be: Media seats that were returned, tickets for the opposing team that were returned, family seats that were returned, or blocks that went unsold for whatever reason.
The ticketing team tells me the best time to buy from the exchange is 1-3 days before the game, and indeed the morning of the game. One rep told there are virtually ALWAYS tickets on the exchange the night and morning before a game, even to playoff games. This is when any remaining blocks are released.
Tickets that appear "out of your criteria" are tickets that someone else has already clicked "buy" on. The exchange was not designed to handle high-demand tickets. When tickets are in high demand, there can be thousands of consumers on the exchange trying to purchase tickets. If, say 4 tickets in a group are available, the tickets become difficult to purchase in a completed transaction, as 4 or 5 people will click "buy" at the same time, and the timer will need to expire, before they are re-listed as "available". I have been told that sets of tickets can live in this limbo state for hours until someone is fast enough to click "buy" and have the server recognize the click and pull the ticket from other buyers. It's one of the most frustrating experiences buying a ticket you could imagine.
The tickets are sold at face, plus a 10% markup for Ticketmaster (who is using it's monopoly to make its nut coming and going). If you don't need a hard ticket, opt for the PDFs, they can easily cancel a barcode and re-issue your PDF ticket at any time if its lost/damaged, etc.
