Is it normal to be 16 and not have had any growth spurts?

Yes & No: By sixteen both sexes should show outward signs of puberty. The individual child may or may not have a noticable change in growth rate depending on how closely you follow it. Puberty can come in quickly over a 2-3 year period or slowly over 5-7. Some kids continue to grow into their early twenties long after others have stopped. A review of their growth chart with your doc should provide answers.

Depends: If the gender is female, then there some concerns that an endocrine dysfunction might be present such as hypothyroidism, growth hormone deficiency or lack of adequate estrogen production. If male, then there might be an element of physiological delay. A bone age will help in this case. Family history if positive for pubertal delay may also reassure.