When you register a domain, you provide the registrar with a list of DNS servers that know how to handle requests for that domain. When an end user wants to connect to one of your services, retrieves the list of these servers from the central registry, selects one of them, and executes the specific query on that server. If you had your DNS split between two different DNS services, you’d have spotty accuracy because you’d have a 50% chance that the server the client queried actually knew about the service you wanted.
There is a workaround, however: you can delegate a subdomain to another set of DNS servers by adding NS records for the subdomain on the parent domain. This would only work if you had, for example, www.yourdomain.com with your registrar and www.do.yourdomain.com with DigitalOcean. If you wanted to have www.yourdomain.com with your registrar and mail.yourdomain.com and intranet.yourdomain.com with DO, this would get very messy as you’d be creating new DNS zones for single hostnames, which is not how the DNS system was designed to work.
All of this said, however, I don’t think DO has a problem with you listing non-DO entries in their DNS zones, as long as you aren’t abusing it (I don’t know this for sure, as I run my own DNS servers). The opposite is also likely true–you could list your DO hosts in your registrar’s DNS.