Can you grab any IP address on the Internet?
On a personal network (LAN) one can simply grab an IP address. If you choose the same IP address as an existing client, you get problems. There are companies like IANA and ICANN that are in charge of IP address bulks and sells them. But what stops you from just grabbing a random IP address? Is this built on trust? What if somebody were to grab an IP address and conflicting would occur. Is there any way of tracing that IP address to the location of the server using it?
Are companies that actually maintain the physical internet cables checking whether or not the clients connecting are using bought IP address blocks?
Solution 1:
There is nothing preventing you from attaching a box configured with someone else's IP address to the internet. However, this won't necessarily cause any issues for anyone else but yourself.
If you steal someone else's IP address outside of the subnet that you are physically connected to, the only thing you will accomplish is not being able to receive any traffic as any router, behaving properly, is going to route traffic to the real owner of this IP address. You might be able to advertise false routes to whatever edge router is upstream from you in the hopes that they would get propagated further in the hopes of getting traffic routed to you based on your stolen IP address, but any marginally competent ISP/upstream provider would never accept routes from its non-enterprise consumers. As far as enterprise customers/other ISPs go, they are bound by specific rules about what routes they can advertise and use with their transit provider or peer, which are monitored 24/7 by Network Operations and Control Teams. Most also have rules about what routes they will accept as valid depending on who advertised them. In short, stealing someone's IP address outside the subnet you are connected to does nothing unless you can also manipulate the upstream routing tables.
That aside if you were to steal an IP address of someone in your same subnet, you would disrupt traffic of both the person who owns it and yourself. With any managed switch or router this will raise alarms as there is a duplicate address on the network and will likely lead to your connection getting blocked in some way.
Solution 2:
Similar to mpez0's answer, but I like a good car analogy...
I'm choosing the UK for this, since that's where I live. Imagine you live in a world where people follow road signs without question, and you happen to live right up in the north of Scotland, about as far from London as you can without crossing water. You live in a small town, and one day you decide that you're going to rename your town 'London' since that will obviously have the effect of driving more trade and tourism to your town, right? You even go to the trouble of updating the local road signs to reflect the new name of your town.
What actually happens? Well, you get lots of people from surrounding villages visiting your town following the signs and wondering why they're not in London. But apart from that, nothing changes. Why?
Consider someone who lives in the middle of England. They know that London is in the south, so when they get on the road to go to London, they follow the signs that say 'THE SOUTH' and keep going until the signs get more specific. In other words, their roadsigns still point to the real London. Their 'routing tables' haven't changed. The fact that your town has decided to change its name to London is inconsequential to them. Their local routes are not nearly specific enough to notice the change.
If you decide to change your IP address, routers elsewhere will not suddenly become aware of this fact. The roadsigns will not change.
Solution 3:
Consider the analogy of your house address. One day you decide to change your address from "123 First Street" to "1600 Pennsylvania Avenue". What difference does it make outside your own property lines? None - because the rest of the world still behaves as if the physical location of your house didn't change, the name of your street didn't change, the city name didn't change, the zip code didn't change... you get the idea.
Mail, packages, and so on will be "routed" to your house based on its location in the address network of your community. The number of your house (computer), by itself, is only the last and final stop (network hop). Everything along the way has to agree, has to synchronize, and you only control the very end of the path.
You can number your house (or computer) anything you want, but for your network traffic to continue to flow you must do so in sync with your environment. To change your house address, you would have to change its number, AND get the bureaucracy to change the name of your street, AND change the name of your city, AND your zip code. Likewise, to change your IP address to something outside your allocated block, you would have to change its number, AND get your upstream provider to change your allocated block, AND change their routers to route that block to you, AND advertise that via BGP to their routing peers.
In both cases you are synchronizing your change with the outside world so the outside world knows how to find you. Otherwise, the effect is that network traffic can't find you anymore - and the only entity affected by your change of address is you. Which, architecturally speaking, is a good thing!