Suppose the average family income in a particular area is \$10,000, will normal approximation give a good estimate for the fraction of families with income over \$50,000?
Given that $\sigma=\$8,000$
I've read it's because income $\ge 0$, I was thinking it was because income may not be a randomly, or uniformly distributed variable?
In a word, no. Incomes are not normally distributed. Here's a graph of the U.S. distribution as it was in 2014, provided by the U.S. Census bureau:
As you can see, this is not at all a normal distribution; it is highly asymmetric. You could conceivably fit a part of a bell curve to part of this data, but there is no reason to think that it would fit any better than, say, a hyperbola, and perhaps it would fit worse.
Curve fitting is not (or should not be) arbitrary. We don't try to fit random curve X to random data Y to see what we get. Instead, we start from a theory about what kind of process could have produced the data, and then we use the theory to predict the shape. Then we might try fitting a curve of that shape to see whether the result supports the theory. Bad fit? Maybe something is wrong with the theory.
For example, consider human heights, which are normally distributed:
Here the model is: a great many factors can impact height one way or the other: heredity (from two parents), childhood nutrition, adolescent nutrition, environmental factors, and so on. When you add together a great many independent factors in this way, they tend to sum to a normal distribution. (This is the central limit theorem).
There's no reason to think that the income distribution is generated in this way, or if there is, the actual data refuses it pretty strongly. You might be on firmer ground if you have a theory that income is controlled by a large number of factors that multiply rather than add. Then you would expect to see the so-called log-normal distribution, which does typically display the sort of right-skewing we see in incomes. (The mean income is far to the right of the median.)
I hope this was some help.