I have arranged my landline phones in PARALLEL with the answering machine so that the answering machine is ALWAYS on and so able to access phone calls. I have put a switch before the phones so I can prevent them from ringing when I do not want to be disturbed. Before I had broadband, this system worked well : I would either answer incoming calls myself, or if the phones were switched off, the answering machine would do it. Since subscribing to broadband, a filter has been added before the answering machines and phones. Now I find that the answering machine no longer takes calls if I have switched off the connection to (just) the phones. It makes no sense to me as the answering machine is NOT switched off, so should take the calls, but it doesn't (when the phones are switched off). I'd like ENLIGHTENMENT about this.More information may help.
Running a switch over a long wire to a common point may violate the common sense rules for telephone installation.  The common sense rule is that the line should be tired to the phone, if erosible. But not in star, but in tree layout.

How many things have you ELIMINATED by trail and ERROR or systematic divide and conqueror.

What kind of switch do your use? There are at least four or more permutations and only one is right. You just can not  just hook up a witch in any way.
 How and where is it wired?
You should understand that the splitter is only rated for one or two sets.Other phones have to come off the line with filters. This implies that you can not have just one switch to turn off the phones, unless each phone has a filter and does not get feed by the splitter.

Does the answer set work when connected to the splitter and no phones on the line?
Do you think the DSL is making the answer set not work right?
Have you TRIED another answer set?

I was not notified about the reply to my post - but can say that, yes, I did try a NEW answering machine last night, and that also failed. It lead to an investigation so I found that one of the phones was faulty - worked in other ways, but prevented the answering machine from picking up when it should.