Car keeps dying, but starts as soon as jump cables are attached ??

I have a 93 Toyota Camry. Runs fine, but
randomly dies on me. One time I was
picking my sister up from school, and I
turned the car off. It wouldn't start again so
I called somebody to come jump it. Right
before they arrived it turned back on. It's
gotten worse. It does it all the time, &
sometimes doesn't start after a while. But
AS SOON AS I get a jump it turns right on.
I don't have to charge the battery for 5-10
min, I just have to connect the cables and
bam. It's on. My friend said it's not the
battery because it turns right on. But the
battery light on the dash comes on so?
Others are telling me it's not the alternator
because if so it wouldn't start at all which
seems odd to me because I had a 2001
Volvo and the alternator was bad but the
car still started. It just died randomly, the
radio would turn off, and sometimes while I
was driving it would turn off. I just replaced
one of the things that hold the battery
wires in place because it was loose and
sometimes when the car died all I had to
do was go under the hood and tighten it &
it would turn on... well now that problem is
solved but the car is still dying... any ideas
?? Alternator?