I would like to purchase a home answering machine that can be set to
pick up after more than 4 rings, preferably after 6 or more.  Other
nice-to-have features would be easy-to-use, digital, fairly
inexpensive, available for purchase in the US, and new (not used). 
(This is for my elderly mother, who can't always get to the phone
before her current answering machine picks up.  I'm concerned she may
fall trying to get to the phone in a hurry.)

You can get a cheap one from AT&T for $19.95 that you can set to
answer from 2 to 7 rings:

http://telephones.att.com/attui/shop/product_detail.cfm?itemID=56c6a95e-8f5b-4cea-9ebc-1098b88bdcd9

I bought a phone from radio shack that answers after 6 rings, built in
answering machine, easy to operate caller ID.  The price was 24.95
www.radioshack.com/cordlessphones

I have a Siemens a/m and fax that I can set to ring as many as nine
times, but it is more expensive and probalby not sold in the States.
Can she pick up after the answering starts or the caller records?
If so, just (I know, that isn't so easy) explain to her that she
doesn't have to rush, that she can still pick up after the phone as
stopped ringing, or get her a phone like that, - and practice with
her, from a cell phone when you are with her.

And try to convince her that anyone who calls, who really knows her,
will leave a message.  Anyone else  - who does not leave a messge - 
will call back, or probably will not.

Thank you all for answering my question so quickly.  I truly
appreciate it!  This is my first time using Google Answers--what a
nice experience!

k1s, the machine you suggested is exactly what I wanted!  I just
ordered it.  I would be happy to pay you for the "find" if you'd like
to ?convert? your comment to the answer.

al1205, thanks for your suggestion, too. (My mom has a pacemaker and
won't use a cordless phone.)

myoarin, thanks for your thoughts.  We've tried to tell her to just
let the machine pick up, but she still tries to get there before the
5th ring.

Thanks again!