The question is if you could actually handle that many lines concurrently, with available software and the hardware resources of a single PC. Rachel's article as well as other comments here indicate that you could not.
From memory, it very much mattered on your choice of serial card/controller. High quality, high speed, (and high price) cards had their own buffers and better drivers resulting in fewer interrupts and more efficient use of host resources when supporting multiple concurrent file transfers at line speed. Being able to drive the modem-to-host connection faster than the modem connect speed was also helpful for modem protocols with compression.
Telephone line modems were always slow compared to the microcomputers of their day. A 14400Baud modem will just send less than some 1.5KB/s. Even a Z80 could handle a few streams of that. I rather suspect this BBS offered fancier services than just chat and file up/download.
You certainly could with later 386 and 486 systems. The multiport cards take care of the interrupt issues. 16 lines, all running at 115200 bps (to allow for better compression!), isn't even 2 megabits/sec. That is worst case. I had a 386 with ethernet and it could push 10 megabits no problem.