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.
If a client doesn't like a design, they can decline that design from the contest. If they don't like the work of a particular designer, they can eliminate that designer from the contest entirely.
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.