For your particular case, it depends on some factors (Hardware and OS).
You have to check the
limits set on your machine (you didn't say anything about the architecture).
Try to check
ulimit -a and take a look, for example, at limits about
open files parameter (it is even relative to the number of open sockets).
These are limits set for user; you can take a look at
/proc/sys/fs/file-max (if you're on an Ubuntu machine) to look at system-wide setting about that parameter.
Other than limits set into the system and limits of your hardware (RAM saturation, etc.), there's no explicit limitation set by SSH Client (as I know).
If you configure limits that you need on your client machine, but you cannot have more SSH connections to the same Server, try to customize followind parameter on the server side
You can customize the parallel sessions allowed from inside your configuration file
/etc/ssh/sshd_config, using as much sessions as you need.
The following parameter should do the trick:
Specifies the maximum number of open shell, login or subsystem
(e.g. sftp) sessions permitted per network connection. Multiple
sessions may be established by clients that support connection
multiplexing. Setting MaxSessions to 1 will effectively disable
session multiplexing, whereas setting it to 0 will prevent all
shell, login and subsystem sessions while still permitting for-
warding. The default is 10.