AT&T BL102-3 DECT 6.0 3-Handset Cordless Phone interfers with the Internet

The first question is the mechanism by which you're getting kicked off, i.e. is it that your computers are losing contact with the Wi-Fi (which is something the DECT frequency bands may help with) or is it that your Modem is losing contact with the other end (likely due to the ring signal is interfering with the DSL connection)?

Once the first ring tone hits, the green Internet light on my modem goes out, the DSL light starts blinking.  Once we are automatically connected again, the DSL light goes steady and then the green Internet light comes on again on the modem.  So I am assuming it's your second scenario.  I do have a DSL splitter on the phone line from the main phone to the wall jacket.

Yeah, those modem symptoms say that the connection on the line is getting torn down, not the Wi-Fi getting interfered with, so it's not the DECT Wireless that's causing this.

"Main phone"?  By this do you mean the single base connected to the lines to which one (or more) cordless handsets attach?  Are there other phones connected to the line itself?

No, no other phones connected to the phone line itself.  The "main" phone is the one connected to the base.  There are two more handsets in two different rooms just connected to the power outlets.

You say you have a DSL splitter. If this is true DSL you need  DSL filter on your phone line at the base unit and from your description that's the problem.

I have it plugged in so that the phone line from the phone base goes into a splitter which then goes into the phone jack.  So am I not doing it correctly already?  Sorry if this is a dumb question.

Like I said, if you have true DSL you need a DSL filter on your cordless phone's base unit. A filter is different than a splitter. If you have a traditional landline and Internet on the same line than you need a filter. You could have a splitter with a filter on one side and no filter on the feed to your DSL modem. 

OK, I am going to show my ignorance here (by the way, I cannot tell you how much I appreciate your help).  First, I thought splitter and filter were one and the same thing.  Obviously not.  What I have (and what several earlier CenturyLink phone guys have left with me) is a couple of what I thought were splitters but maybe they are a combo splitter/filter??  These have one "male" outlet that goes into the phone jack in the wall.  Then they have two "female" outlets - one marked "DSL" and one marked "phone." Then I also have two with only one "male" outlet and one "female" outlet. I have been using the former with the base phone line plugged into the splitter side which is marked "phone." I am using one of the splitter/filer thingy with our Samsung Smart TV which is the type that only has the "male" and one "female" outlet.

I have been told by CenturyLink (when they installed the DSL line) that I have one designated DSL line that comes into the house.  This line (green) is connected to the back of my modem.

As a matter of information, I also have a fax machine which has is connected to the phone "female" outlet of the dual splitter/filter and then into a phone jack in the wall.

Do I have them connected correctly?  Just for kicks, I tried plugging the base AT&T cordless phone directly into the modem and used my cell phone to call "home."  However, I still got kicked off the Internet.

The one marked DSL goes to the modem, the one marked phone is the filtered side, this would be used if the phone and DSL service were from the same wall jack. The line to the modem and green is the color used for identification purposes for self-installs, it just identifies it as an unfiltered line. All other equipment, phone fax or whatever needs to be filtered. If it doesn't say it's a DSL filter than it isn't. Your splitter that are labeled DSL and phone the phone would be filtered. There's always the possibility you have a bad filter. Unplug everything that requires a phone line, not the modem line, and hook up one at a time, try a call if the modem stays up that one's ok, do it with all that should be filtered to find the cause. The one to the TV wouldn't be a phone type cord it would be an 8pin ethernet cable that would come from an ethernet port out of the back of the modem to the TV, so if you have a regular telephone cord going to it, the female jack on the TV would be too big for the cord. Take everything one step at a time.