Can I run traces through "No Connection" pins?

It's a grey area. Some manufacturers tell you that's used for calibrating. Some manufacturers even will tell that you that certain pin has a function, only used by them for calibrating. Some tell you only not to connect it, or just say it's an unconnected pin.. You can't know for sure. The datasheet is information the manufacturer want to tell to you about using the device, but it might not be everything.

I recommended you do not connect them. If you get some generic IC from a different manufacturer or even batch the behaviour might be different. If you're engineering a project, you don't want to throw in unpredictability. You would have to test every single batch before you're going to use that particular batch. It depends on whatever you want to do that.

Normally "n.c." means that the pin is not connected to the die, and you should be safe running a trace over it. In some rare occasions I've read "n.c. DO NOT CONNECT", which rises the question why I shouldn't, if it's not connected internally anyway. In any case I expect the datasheet to mention it expressly when I shouldn't connect to the pin.

One example is pin 4 on the LP2981. Texas Instruments says "Pin 4 (NC) must be left open. Do not connect anything to this pin", without further explanation. National specifies: "Post package trim. **do not** connect to this pin". So the pin seems to be connected as expected, otherwise it would be safe to connect. In this case "not connected" should be read as 'no end user connection". Both manufacturers indicate clearly not to connect.

If you must do it, at least check an actual device with a meter - check resistance between the pin and +ve and the pin and ground using both polarites on the meter - if this shows infinite resistance it probably isn't bonded and is safe to connect.