1. Rounding errors. Pie, for example, has been calculated out to over a million points beyond the decimal yet it never hits, "00000000000000000000000000000000000000000....." Every calculation using pie is wrong, at least technically (I don't know how to type the symbol for pie). As is every calculation in which you round.
2. Even a hand Calculator has a compiler that translates formulas from base 10 to base 2. The compiler could be programmed wrong, especially in a hand held calculator that costs a dollar.
3. The machinery that stamps out the logic boards could have a flaw, probably one that developed over time. If one circuit that was supposed to be closed, was open, or Viceversa, it would create errors.
4. Entering the numbers wrong is the mistake I often make.
5. Putting a decimal point in the wrong place. The movie, "Office Space," implies that this is the source of their error.
6. When dealing with numbers in the billions and trillions, and multiplication, the calculator simply can't handle that large of a product. To get around the problem, you can drop zeros from each number, but you have to remember to add the correct number of zeros back to the product. Adding the wrong number, or forgetting to add any, is an easy error.
7. When using more complex geometric functions, like sine, tangent, arc-sine, and so on, just because the calculators will calculate them doesn't mean the user knows how to interpret them (misinterpreting results).
8. Hitting plus instead of minus, or divided by, instead of times. Reversing the numerator and denominator, on and on.
9. You drop it and the battery pops out. You put it back in, but when the exam begins, you turn it on and nothing happens. This would be the error of errors. If you were doing the calculations by hand, this would never be a problem, and I suspect this is your teachers point - to illustrate some of the potential problems when you turn something over to a machine.
10. You have a calculator powered by light, but the test is in a very dimly lighted room and the calculator won't work.