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On a generic question like this only a generic answer can be given. The amount of traffic on a network becomes an issue if the services provided over this network are impaired such that they cannot provide required the quality.
How this plays out in real life depends on the network traffic characteristics and services provided. For instance high jitter on a VoIP network may impede the quality of voice calls. But this could be mitigated by proper QoS provisioning, so does not necessarily have to cause problems with the voice service. Payload fragmentation may cause connection bandwidth restriction. But proper MTU management may prevent fragmentation. As you can see numerous factors come into play.
