The short answer is NO.
Desktop versions of Windows do limit simultaneous inbound TCP connections (license wise and/or technically).
Specifically, Windows XP / Vista limits this to 10 connections, and Windows 7 / 8 / 8.1 / 10 / 11 limits it to 20 connections.
However, most DNS requests are not sent via TCP. They are sent via UDP, which is a connectionless protocol, and therefore cannot really be limited.
DNS generally only uses TCP connections for zone transfers. So this limitation would only be a problem if you have more than 10/20 secondary DNS servers all trying to connect at the same time.