No, there is no limit to how many designs a designer can submit to a contest. Nor is there a limit the total number of designs a contest can receive.
But we have set some initial design limits, which is extendable, that also depends on the package chosen by the customer while posting the project. In only fast track contests there is a design submission limit per designer of 4 designs per contest. In all other packages, designers are allowed to submit as many designs as they want in a contest.
Generally, the number of designs a contest receives will depend on the quality of the visual design brief and the amount of feedback the client provides for designers. Most designers prefer to enter contests where the client has left plenty of feedback for the participants, and written thorough, clear, and concise visual design briefs.