What's the maximum number of wifi connections for a single WiFi router?
While Oneiroi's answer starts to get you there. There are a few flaws that deserved more than a comment on his answer.
First, the bandwidth you are actually contending for isn't the back end ethernet link, but the wireless link speed. So on a 54 mbps router you would be contending for that 54 mbps. At 60 clients that would be about 900kbps each, not counting TCP overhead, counting TCP overhead you are already down to ~720kbps.
Secondly, you still have to deal with signal strength issues, which depending on your location could greatly effect throughput.
Thirdly, you have to deal with interference between those 60 radios all trying to broadcast. Years ago when I was working on planning an 802.11b wireless network the engineer we talked to said the rule of thumb was about 10-12 clients per AP for best performance, you can probably move that up to 20-25 (pure off the cuff number)with today's newer technologies. But that still doesn't get you to 60.
I would suggest that you take a look at some pure Wireless APs (not routers) and do a site survey to find the optimal placements for them.
Concurrent connections are dictated by 2 things, bandwidth & resource,
On the resource front it would be best to ask the vendor, the cpu & memory of he hardware could be a limiting factor in the supported concurrent connections.
Bandwidth is the major limiting factor, whilst you may connect @ 54mb/s remember if you users are contenting with 50 others all over the 100mb/s Ethernet attached to the access point their actual available bandwidth becomes 2mb/s.
As you start to reach capacity you will find:
Bandwidth: The Ethernet connection is always transferring the maximum available for the connections (i.e. 100mb.s)
Resource: assuming someone isn't attempting to crack your network by spamming de-auth packets you will see a lot of "broadcasts" from the AP, as it tries to reconnect with dropped clients.
I don't see that model specifically, but similar model lines look like they're a SOHO class routers. The price on those models makes them look like a step above home routers, but I'd be wary of their limits.
If you're pushing 60+ devices through a wireless system simultaneously, I'd really strongly consider getting a dedicated access point and a separate router. Combo router/switch/wireless AP's in my experience tend to get glitchy and have issues with maintaining connections or having a subcomponent die and require a reboot, killing everything until it restarts.
They also tend to have limits in their routing tables that start giving odd behavior for things like web browsing or computers that get booted off the network.
If I had to guess, you're overpushing that router. Suggest getting separate, higher-end systems.
As others have said, you're probably "oversubscribing" the wifi router. Wifi is a shared medium so all 60 clients are sharing and contending for the 54Mbps (assumed) bandwidth available through the wifi link. That's somewhat analogous to connecting 60 wired clients to a hub and uplinking that via a single 10, 100, or 1000Mbps link to your core switch, but in addition you've also got signal quality and potential interference issues to deal with as well.
I should add that bandwidth is not an important parameter for consideration. It is obvious to divide bandwidth on number of clients but real hardware has various limitations.
Like count of hardware encryption modules (it can be around 32 or 50/64). See https://superuser.com/a/639007/49345
Manufacturer for home end products like D-Link recommends only 15 simultaneous connections.
NETGEAR home routers can accommodate up to 32 clients per wireless band. If your router is dual band, the total wireless clients your router can handle is 64 (32 for the 2.4GHz and 32 for the 5GHz)
Statistics on Cisco 1142 and 3502I access points:
Protocol / bandwidth / # of Clients per AP 802.11b 600 kbps 13 802.11g 600 kbps 43 802.11n (1 Spatial Stream) 600 kbps 43 802.11n (2 Spatial Streams) 600 kbps 273
and they tell about 30-40 clients per AP.