Vegetables are tender when the point of a knife penetrates them without resistance. This technique yields tender, not crunchy, vegetables. In most cases you should support the vegetable with a fork or spoon while inserting your knife. Potatoes tested this way will cling to the blade until fully cooked, at which point they will slide off.
For those who prefer undercooked, crunchy vegetables, biting into one is the best way to check for doneness. Lift a string bean or broccoli stem from the pot, run it under cold water, and taste it. The difference between perfectly cooked and overcooked vegetables may be just a matter of seconds, so pay close attention while cooking them.