Here's the short answer: Yes, I think you can be a vegetarian and Paleo at the same time, depending on what you call "vegetarian", however, you can not be vegan and be Paleo.
Let's start from the end.
If you're vegan, you probably eat a lot of soy, beans, grains, vegetables and fruits and hope. Probably a bit 'of nuts and seeds. People need protein, and all these things except for the fruits and vegetables provide a measurable amount of protein.
For comparison, I would like to point out that 100 grams of beef steak gives you about 29g of protein, and 100 grams of chickpeas they give you about 8 g.
Many experts have said that for optimal health and wellness, people need between 0,7-1g of protein per kilogram of body mass per day.
In this case, I personally would need to eat more than 2 pounds diceci every day to balance the recommended dose of protein. O 1kg. almonds, or 23 slices of whole wheat bread, or 450g.di tempeh (fermented soybeans).
So, if you take away the soy, grains, beans, and more to make your Paleo diet, were nuts and seeds as the main sources of protein, and I said that I should eat 1 pound of almonds a day to meet the my needs.
So you can be vegan and Paleo? Well, you would have resulted in an increase in weight, caused by the need to get a good amount of protein, so my answer is no.
Here is an account of a former vegan converted to Paleo, describing what you can do as a vegan to make your diet better without consuming animal products.
"As for vegetarians, we need to understand who is a vegetarian. I never understood why one is considered "vegetarian" and continue to eat fish, eggs or poultry at times. The last time I checked, salmon and chickens were not among the vegetables ... However, if you're that kind of "vegetarian", then you can live well with the paleo. You should try to eat more poultry, fish, and if you do not have problems of consumption of eggs, also eat more of those.
If you are a vegetarian and the only source of animal food that you eat are dairy products, so try to buy them at zero distance or from a local producer and always full. Sarai Paleo if you do so? No, and I do not encourage people to eat dairy products.
If you eat dairy and follow all other Paleo principles, you will be "Primal".
Primal is close enough to the Paleo, but with the addition of a high-fat, high-quality, preferably fermented milk if you can digest it.
One cup of whole milk still only provides about 8 grams of protein, but it is certainly better than nothing, if you do not eat soy, beans, and other grains as a protein source.
in which case you may not be Paleo.
In addition to this, if you want to become vegetarian or vegan diet, you do these things. Avoid vegetable oils such as soybean, corn, cotton, and other oli treated high in omega 6. Do not drink your calories in the form of sugary drinks or juices. Avoid refined sugar, including high fructose corn syrup, white cane sugar, and many others. Do not consume food, obviously false, as aspartame, other fake sugars, food coloring. Do not rely on caffeine to keep you awake during the day. Also, I recommend a good multivitamin so you do not become deficient in substances that are found only in the flesh. "