It completely depends on your configuration and use.
Generally as user demand grows over time a company will add resources (servers, memory, faster database, key-value stores) to ensure that user response time stay within what they determine (or allow) to be acceptable limits. When front-end revenue starts to drop this is a big place to look.
So in the short term, yes one server getting increasingly load will at some point start to slow down. But on some servers that could start to show at 10 users and others not until 1000 users. It depends on threads, memory, garbage collection, what the code is and does and many other factors.
Overall a company should determine what it wishes its users performance to be and then provides the necessary infrastructure to support that as usage grows. With auto-scaling in the cloud this is increasingly easy.
Also - make sure that you are testing a production or production-like system to have any meaningful conclusions about performance. Also make sure that your jmeter threads provide enough similarity (i.e. variety) in content to real requests on the web.