Some types of firewall, and almost any router doing Network Address Translation (NAT, also known as IP masquerading), will forget about a connection through them if the connection does nothing for too long. This will cause the connection to be rudely cut off when contact is resumed.
You can try to combat this by telling PuTTY to send keepalives: packets of data which have no effect on the actual session, but which reassure the router or firewall that the network connection is still active and worth remembering about.
Keepalives don't solve everything, unfortunately; although they cause greater robustness against this sort of router, they can also cause a loss of robustness against network dropouts. See section 4.13.1 in the documentation for more discussion of this.