In my experience, about 3 to 4 weeks maximum. The colder the refrigerator is set, short of freezing them, of course, the better. It helps, too, to keep any overripe produce such as very ripe apples out of the fridge, since these emit ethylene gas, a plant hormone which promotes ripening.

You might try putting them in those 'Debby Meyer Green Bags' -- a special plastic bag designed to extend the life of produce. I guess you can buy these online or from several stores. I've heard they actually work quite well, extend the life of stored produce a long time, but I haven't tried them myself.

Bartlett is a "summer" pear, I guess, and isn't designed for long-term storage. At least that has been my experience. In contrast, my Bosc pears and my Korean Giant Asian Pears, both of which don't ripen until mid to late October, will keep in the refrigerator for me until at least February or March most years.

I know commercialy Bartletts and other pears are kept for many months, but they are placed in special conditions -- one thing is that they are kept in an atmosphere that has little oxygen, which prevents further ripening. Also, I seem to recall from my Intro To Horticulture class years ago at Michigan State that apples and pears are held at 31 degrees F (-1 C) to promote maximum storage life.