What does the soil look like? What direction is the window facing?

It faces south, but there is an overhang, so it gets a little bit of direct light but not much. The soil is standard potting soil from the store, somewhat coarse grained. The water here in LV is generally high pH and high in calcium.

It looks to me like you may be under watering and it has too much fertilizer salts in the soil, this could be caused by water with a high PH or fluoride in the water. They are very susceptible to fluoride poisoning. You should probably flush the soil. Most houseplants with the exception of cacti like the PH to be below 6.5. Take a hose and flush the soil for about 10 minutes or put it in the shower and let the water soak the soil. Let the pot drain and put it back where it goes.

I would then start using distilled or rain water to water it. 1 cup a week doesn't seem like it would be enough for a plant that size either. You should be watering enough to flush the soil of salts.

Also do a search of the forums. I'm pretty sure this question has been answered hundreds of times through the 20 or so years GW has been going. You may glean something that I've forgot.

If it's foliage burn from mineral salts there should be a white salt deposit on the edge of the dead parts where they meet the still green parts of the leaves.

No mineral salt deposits form what I see. I'm going to start giving it distilled water from now on, see if it makes a difference.

How long since it was last repotted?

It was in a very small pot when I got it from the store last spring, and I moved it to a bigger pot some time around then. That's when it started going brown at the tips. I didn't prune the roots or anything, just moved it to a bigger pot.

"Repotting vs. potting up - that there is a difference between the two
operations is a concept foreign to most hobby growers. One of the
practices ensures your plants will at least have the opportunity to grow
to their genetic potential within the limits of other cultural
conditions; that would be repotting, with its accompanying root
maintenance, complete or partial bare-rooting, and a change of soil.
Potting up, on the other hand, only temporarily allows the plant to grow
a little closer to its genetic potential before root congestion and a
diminished number of fine roots quickly returns the plant to the state
of limited growth and vitality it was experiencing before potting up."

- http://forums2.gardenweb.com/discussions/1490818/good-growing-practices-an-overview-for-beginners?n=143

Don't give it a cup of water per week. Water it thoroughly so water flushes and drains out of the bottom. Do this when the top inch or two of soil feel dry (stick your fingers in). Never let the pot sit in water that has drain out. Always empty the saucer too.