Yes, you can, but the improvement in your fit will be compromised.
The bite of your new denture with the old denture depends on how worn and ill-fitting the complimenting denture is. For example, if you have a set of dentures that are 5 years or older and we replace only one of the dentures, then we will need to take new sharper teeth and grind them down to sufficiently mesh with the older denture, because the teeth on the older denture will be worn down.
The best advice is to replace them as a pair. This way both the upper and lower teeth are sharp and new, and they are constructed to mesh precisely with each other. This will allow for a better fit, and provide more ease in chewing your food.