How Many People Can Login To SQL Server With The Same User Account?

I'm developing a client-server application using Visual Basic and SQL Server
6.5 for 700 users. The application currently requires users to login to the
database using their network account (via trusted connections). Each of the
users takes up a maximum of 6 connections. Unfortunately, other
considerations will force us to discard use of trusted connections. So it
only leaves us either to create 700 user accounts on SQL Server separately
from their network logins or to create one SQL Server account and everybody
uses that same account to login to the database. For political reasons, the
customer would prefer not to ask their support group to maintain 700 user
accounts on SQL Server if it can be avoided.

My question: is there any technical limitation or other negative consequence
of having 700 users login to the same account to SQL Server if SQL Server
can handle that many connections (it would be 6*700 = 4200 connections)
simultaneously? Are we supposed to do things like this? It seems that we
don’t have better choice than this.