Use iPhone for internet and keep access to home network
In theory, it is possible. Practically, it's not worth the bother for most, if not all, home users.

In order to fully combine the bandwidth (i.e. without having restrictions on single connections), you would need to have a "partner" server on the Internet that you can communicate with trough your local ISP and through your cell connection. That partner proxies all traffic to/from your network - essentially combining the data from both internet connections and sending it on as one.

For that to work you also need a specialized network device on your home network that is placed in between your local network and your router and the cellular connection. It splits up traffic and sends it on to the proxy server (and vice versa for incoming traffic from the proxy server). 

All in all, it requires specialized knowledge about networks to set this up, it requires additional equipment on your home network, and it requires a server on the internet. This is usually relatively costly to come by, so it's not really worth it to add a smaller amount of extra bandwidth to a home user connection.

Larger companies (as well as smaller companies specializing in something that requires specialized network access) implement something somewhat similar by the way of BGP routing. This allows having multiple paths to the Internet, and to be reachable through multiple paths at the same time. However, it also requires you to have your own autonomous system number (AS-number). It is free, but takes time to apply for. As well as requiring your to have your own set of routable IP-addresses (usually PI for smaller setups, and PA for larger). These IPs are getting harder to come by. Usually you would need to have at least a /24 (i.e. 256 IP-addresses) in order for this to work at all - probably even more today. If you have a use case where IPv6 only is enough, the requirements are a lot easier to fulfill. However you still need to have corporation from both ISPs in order to make it work.