Definitely possible. My connection to 3 mayflower people (4 if you count the father & son team of Francis and John Cooke) goes through my Perry relatives. I submitted and was approved on pilgrim Edward Doty. I could submit on the Cookes and Richard Warren but it costs for each application. Gathering all of the records for the application was not easy but well worth it.
There is a set of books called Mayflower Families through five generations (nicknamed the Silver books) that are published by the General Society of Mayflower Descendants. They document 5 generations down from the pilgrims. You will have to prove your lineage up to where the books leave off. Good luck!