Can I have PayPal submission and Pay by check on the same form?

Yes, you can do that. Here are the guides that might help you with what you would like to happen: 

How-to-set-up-paypal-along-with-other-payment-options

How-to-Create-a-Basic-Multiple-Payment-form

Should you be having any questions or you need clarifications and/or assistance with those guides, please feel free to let us know and we will be glad to help you any time. Thank you.

So While I waited for the answer, I created a second Submit button.  The first one is tied to the PayPal payment options, the second one I renamed Pay by Check and it just submits the form. It seems to work.  What type of problems can this cause?

That will be confusing to have two submit buttons, and they will actually do the same, which is submit the form, and if the customer has selected any product from the payment integration with Paypal, the form will redirect to Paypal page no matter which button was hit.

So I do recommend you to follow any of the guides provided by my colleague.

Willy, as we can only support with a single question per thread, your message has been moved to a thread of its own, here:

http://www.jotform.com/answers/731739

We will be supporting you with your question there, shortly. :) Thank you.