If you're 2.5 miles from the "home office" (typically an MDF/IDF - main / intermediary data facility), you are outside the "sweet-zone" of ADSL.
Typically, (A)DSL sees maximum throughput at 2 miles or less from a MDF/IDF. Basically, the distance between your demarcation point (modem) and your ISP's closet.
You're only half-a-mile outside the sweet-zone, so I would still expect you to see reasonable speeds. That said, you might never see a stable 6Mbps, due to ADSL typically using older, buried phone / Cat3 lines. (Unless CenturyLink ran shielded coax/twinax copper or fiber, which I highly, highly doubt.)
The burst you see with TMN is likely because the lines are cool, so there's no electromagnetic field around them. Because ADSL is typically Cat3, it's often minimally (if at all) twisted, unshielded, and poor-cabling, so it's easy for the cabling to build an electromagnetic field that interferes with signal transmission. This field won't exist at an idle, it will only exist when data is being transmitted. Once the field has built up (often as quick as 50-2500ns) it starts creating cross-talk and signaling interference, slowing the throughput you can achieve. (Basically, error-rates go up, and as a result more of your bandwidth is spent on handling those errors.)
Unfortunately, without replacing the cabling, your speeds probably can't be improved much.
All that said, does CenturyLink offer 5Mbps? If so, I would downgrade to that and run some more testing. If their signaling is wonky, you'll see a similar drop (I would expect either 1Mbps or 833kbps drop when you switch if they have a signaling calculation incorrect). If you don't see a drop, save yourself the money and keep the 5Mbps connection. If you see a drop, document it (now, the 5 vs. 6, and then, the x vs. 5). If you still have a drop, call CenturyLink and explain the situation. I have no idea how well they'll assist you, but they should be able to make other accommodations (provide 6Mbps but bill at 5Mbps, for example).