More like your ISP will run out of IPs to give its customers. Which will mean there will be a definite upper limit to the number of customers a given ISP will be able to support at any given time.

More likely, though, is it will force your ISP to reorganize their IP space more efficiently, or, put newer customers on an IPv6 network (which will be the "IPv6 Internet," not the "IPv4 Internet" everyone knows and loves.