There have been times when I was without a sharpener, and a pencil I was using became dull. I tried using my knife, but that seems to break the point. Am I doing it wrong, or is there a better way to get out of the problem? I was using pencils specifically because they're erasable.
You're doing it wrong. Most artists I know use a pen knife to sharpen pencils. The trick is to only bear down on the wooden part and cut along a shallow angle. To finish the graphite point, use extremely light pressure.
I don't carry a knife. I have often been drawing/sketching while in a park or on a city bench near a curbside and I have found the rough pavement and/or concrete on the ground works well in a pinch (on those occasions when I don't have my sharpener with me). You're not going to get a perfect sharp tip, but it will remove the wood enough to get more of the graphite out and available for continued use.
Similar, if you are near a brick building, you can use the mortar in the joints between the bricks as it provides a nice channel for the pencil to shimmy through.
I like to use an emory board to sharpen or shape a drawing pencil. I don't expect it would work well for starting from scratch ; ) with a new pencil, but works well otherwise.
It is actually easier to sharpen a pencil with a box-cutter than a pocket knife (it is sharper, so you need to apply less pressure, which allows you to have more control over what and how you are cutting). Because using a knife for sharpening takes some practice, it may be a good idea to use a box-cutter until you get used to it.
Another option is to use a mechanical pencil instead - no need for sharpening.
A rough grit sandpaper works in a pinch, as do nail files and other wood or steel files.