I would have a urinalysis performed on her urine. If she has a urinary tract infection I would also perform a culture and sensitivity on the urine to see which antibiotic would be the most effective. This problem could also be a sign of early kidney disease, which can be determined by checking the microalbumin content of the urine (usually a part of a complete urinalysis). If microalbumin is found in the urine than blood work is warrented to determine the extent of the kidney disease. A CBC/Chemistry/T4 should be performed. This will determine if there is diabetes, anemia, kidney disease, liver disease, and hyperthyroid disease,etc. Incidentally, a 16 year old cat is a senior citizen and should be checked for all of the above anyhow.
Don't assume that the inappropriate urination is just due to behavioral causes until all the physical causes are eliminated. If all tests are normal than it is probably behavioral, and if so, you must try and discover if there is any change in the enviromment that has effected her adversely such as a new pet or a new human in the enviroment, work men in the enviroment, changing the furniture around, etc,etc, Whether something has changed in the enviroment or not, purchase some Feliway to use to try to make her comfortable again. Feliway is a natural feline pheramone in a spray or air-freshener form. Use as directed. Sometimes this product is enough to get a cat back on tract.
If Feliway does not work there are several different medications that can be tried, such as anti-anxiety medications, anti-depressents and others. It may take one or two course of medications to find the one that is effective for her. Just try to be patient because in 99% of the cases of inappropriate urination there is an effective treatment.
Are the messes feces as well as urine? My elderly dog lost bowel control with the onset of canine arthritis and sometimes does not know when a movement is on the way to such an extent that she messes her bed. I don't know if the same goes for cats but you might want to ask your vet. An x-ray was the only way I found out; maybe you could just watch for apparent weakness. If there is arthritis, it might be painful for the cat to climb into the box. I live by "Nature's Miracle" these days.