Have you contacted your neighbors and and asked them what modem they use? Do they confirm what the Tech tells you, that they have 3.0 Mbps download speed?
Read the reviews of a billion 7800n router on www.amazon.co.uk They are designed for long ADSL lines and are popular in UK and Australia. You can change the signal to noise ratio on the line to get higher connection speed.
Turns out that CenturyTel Support now says ... I checked and it looks like 3 Mbps is already exhausted, it means that is full of users, that's why you are not able to use this speed at this moment I do apologize for it because I know your neighbors enjoy that speed but unfortunately is not available at the moment.
I had ADSL modems in past and it's all about your phone line quality and ability. For example I still was not able to manage something like ADSL2+ or so cause my line simply didn't allowed me to do that. So I think here lies your issue no matter your ISP, ask phone company about that.