Yes, you can be sensitive or allergic to nearly any food and you can develop that sensitivity at any point in your life; you don't have to be born with it. Some food allergies/sensitivities are more common than others, though.
Having said that, are you sure it's the spinach? Are you eating it plain with nothing else added?
With some allergies, cooking the food in question seems to make a difference....it does with my DH. He can't eat apples, peaches, celery or carrots raw, but he has no problem with them cooked because cooking denatures the protein molecule in those foods that he is allergic to.