For short periods of time, I think yes, by relying on roots and fruit. I know people who have done this. But long-term the deficiencies will catch up to you unless you supplement.
Nutrition and Physical Regeneration has a post about Orthodox Christian fasting, which is mostly vegan with some shellfish allowed. I have met a few Orthodox paleos who manage this. Michael isn't paleo and uses lentils. Paleos have to rely on SOOOOOOOOOOOOOOOOOOOOOOOO much coconut it's ridiculous- like 2-3 cans a day. That plus starch allows one to be OK and maybe even thrive, but it's very difficult to get all that food down the hatch. I've been palel-pescatarian before and you really do have to force yourself to eat a lot of very satiating starch and fat, which can be hard because it's so filling, but if you feel full and haven't gotten enough calories you will feel kind of sick later.