Did they check the alternator, has this been happening for a while
because if the battery wasn`t charging soon it would be dead and not
start. It could be the fuel system or alot of other things not related
to the battery. Ive bought new batteries that went bad in a few months.

Agree with Mark, it could well be something else..once an engine comes down
to less than 200 rpm or so (as indicated on the tachometer in your
instrument panel) it will probably stop (stall) alto-gether.

Somethings a miss in the engine management or the idle control area. Take it
to a good mechanic, not necessarily Toyota,..just a reputable mechanic.

Jason

Hmmmm....two thoughts.

Check the connection at the battery. I had an old Tercel wagon that would
always start on the 1st turn of the key, and after replacing the battery I
hqad a problem like yours. i looked and looked and then finally noticed a
wire that had broken off the harness connecting to the battery.

Also, you say it was in an accident. I would be looking at the wiring to
the fuel pump, althought this waoula also cause the car not to start at
all sometimes, or to conk out while driving at speed...