There's not a hard limit to Server per-se, but there can be a limit depending on the environment. Also, the number of services isn't really the question - the issue is the number of service instances that are spun up (default is Min 1 Max 2). That's why you can publish and start more services if you stop some of the existing ones.
There's two ways to approach this situation. One is to alter your service design, the other is to alter the environment.
I'm not sure what your end goal is, but personally, in terms of organization, I'd create a map document, say, for 'birds'. In that map document, I'd create sublayers for different species of birds. Then in your application, you can reference the individual feature layers as you need them. You may already be doing something similar, and this will result in a lower number of service instances running on your machine. I'd then have services for 'birds', 'monkeys', 'snakes', or whatever.
Outside of map design, I've seen two distinct environment issues come into play when publishing a large number of services to ArcGIS for Server.
First, I'd check your virtual memory settings - ensure it's set to 'Windows managed' Hamstrung virtual memory can affect the amount of physical memory Windows can allocate to processes.
Sedcond, you may also want to check out this KB, it discusses the impact of the non-interactive desktop heap has on the number of services you can publish to a Windows machine.
Finally, building up your stack and leveraging Portal for ArcGIS along with the ArcGIS Data Store will allow you to publish potentially thousands of feature layers without the overhead that ArcObjects based map services bring.
Hope that helps!