So look at the symptoms and what causes them...
All phones lose connection... look at things common to all phones. Router, ISP/Uplink, core switch. You must have at least 3 switches at this location, are they daisy-chained, or is each one connected individually to a router or master switch port?
Repeatedly cycles through power-up sequence? This is caused by power issue or connectivity... If they stay at DHCP and just time-out, they are not getting a DHCP response from the router. Check your DHCP lease time and pool size... 157 phones with no vlans implies there could be at least 157 other network devices, almost 300? If so, this network should absolutely have VLANs and unique broadcast domains... Still, if it's a flat network verify your pool size is big enough because a /24 network won't cut it, and if it's a tight pool size, lower your lease time to 4 hours or something.
When you say there is no response to power cycling the phone, you mean it isn't resetting or it isn't working? if you power cycle the phone and it doesn't fix the problem but it just works later, this tells you the problem is NOT the phone... But a connectivity issue.
Recovery time of hours to minutes isn't very helpful in this case.
My guess is one of a few things... DHCP (as noted above), a bad patch cable that is common to all devices. or a defective switch... Since you implied that all devices do this at once, start at the most common devices. Although the problem can start at a device and spread throughout the network, it usually doesn't, but I would be lying if I said I had never seen that happen before. Connect a PC with Wireshark capturing data... You don't have to a packet master to use wireshark, just look for anything unusual when this occurs. I ran into a site where a smart TV in the conference room was flooding the network at random times, basically grinding everything to a halt then it would just start again. Nobody even knew the TV hanging on the wall was plugged into the network, but a firmware update later and haven't been back to that site in over year.
And now I am rambling... Need more coffee...