I'm going to keep this overly simplistic because I don't know how much you or someone else reading this understands networkworking, but....
No, that's not the way networking works. For one thing, the IP address to your laptop is determined by the device it is connected to (switch/router). Individual network devices are setup to talk to a certain range of addresses and their connected devices must be in that range in order for them to be able to communicate.
For another, most networks provide addresses to devices as they connect (called DHCP), and reuse those addresses as devices drop off and others come along. This is especially important for places like coffee shops where a large number of random devices will connect with the network there.
Another important note is that in places like coffee shops, you need to let the router determine the address because if someone else with a laptop with the address 1.1.1.1 (or whatever you have) is already there, you won't be able to connect due to there being an IP conflict.
Also, as mentioned by @HazardousGlitch, the IP address your corporate system would see from a coffee shop, or even at home would be the modem's address, not your laptop. So if you are at a place with 10 computers, the corporate network would see just the place's modem, which would then re-address incoming mail to the approriate computer.
It is possible to have a static address at home since normally the devices there don't change much, but it usually isn't efficient and won't fix the external modem address mentioned above.
EDIT: Upon consideration I thought I should add the most important reason adding a static IP address won't let the user connect to his corporate network, which is because any competent IT department won't use that as a consideration for connecting in the first place. If all it takes is having a certain address, then all anybosy has to do is configure their device with that address and viola! They're in! Organizations will allow authorized users to connect through a variety of methods and controls such as internal VPNs, Active Directory, or MAC addresses to determine who can connect from outside the workplace.