If you’re getting the error “you’ve reached the maximum limit of customers that can be assigned at one time,” you’ve reached the maximum number of Messaging Customers that can be assigned to you as configured in the Messaging Routing setting. You won’t be able to receive additional Messaging and Email Contacts even if there are some waiting. This applies specifically to Messaging (i.e., chat, social messaging) Channel only. You must close Messaging Conversations to go below the maximum capacity allowed, or, an Administrator can increase the number of maximum Messaging Conversations allowed to be assigned on any given time.
If you made yourself available for more than one Channel, like Voice and Messaging Channel, you’d be routed Voice then Messaging Customers first (ahead of Customers reaching out over Email, Task, Voicemail Channels). You won’t be routed emails or voicemails until there are no Customers waiting for Voice and you’re under maximum capacity for the Messaging Channel.