Does the number of nodes determines the rate at which a neural signal is transmitted?
The number of nodes is a trade-off between speed of conduction and reliability. The fastest transmission would be where nodes are spaced at roughly the maximum distance to trigger sufficient depolarization at the next node - however, small changes in ionic concentrations or channel densities could lead to failures, so in practice the nodes are typically a bit closer.

The slowest transmission would be to have no saltatory conduction at all.

(I mean these to be general principles; one could probably design a hypothetical system with very close nodes that is actually worse than having no saltatory conduction, and there is probably a perfect edge case where nodes that are spaced at the maximum distance are slower because of membrane time constants)

The reasons for this are: that travel between nodes is very fast, regeneration at nodes is comparatively slow. However, there is a maximum distance for reliable transmission, so we can't simply have one long stretch containing no nodes in between.

edit: references were requested. This was off the top of my head so I would suggest any introductory neuroscience textbook, Purves is a preferred one of mine, but this will run you significant $ if you are not/have not been a student in a neuroscience course. The Wikipedia articles for https://en.wikipedia.org/wiki/Node_of_Ranvier and also for Saltatory Conduction both look good to me, though they do not address the specific issue in detail. This brief Neuron review is more appropriate to the specific question asked, though I am not sure people without an academic library subscription will have free access: http://www.sciencedirect.com/science/article/pii/S0896627300803232