Flight Search Engines And The Multi-City Ripoff
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Booking those segments separately means that you aren't just taking what the other airlines report as the fare but it is looking for the lowest possible fare on each airline, issuing a ticket on their own ticket stock. For example, American and United aren't partners so I can book an United leg on an American ticket but even if there are supercheap fares available, it's going to give me a relatively high fare for the "foreign" leg.
In general, what Adam Goldstein said about the downsides of buying a bunch of one-way tickets is correct but the baggage issue depends on the carrier. Most of the big carriers have interline baggage agreements with the other majors eliminating the need to retrieve baggage and recheck it. So if I have a Delta flight and a United flight on separate tickets, I can tell the Delta agent at check-in to check my bags through to United. You would do that by showing them an itinerary or proof of a ticket. Low cost carriers like Southwest, Spirit, Frontier, and Ryan Air generally don't have interline baggage agreements with anyone.
Where this breaks down is if you have irregular operations on one of your flights. If the inbound is late and you just make the connecting flight, your bags probably won't. You'll also get the run around if your baggage gets damaged. Generally I'd build in a 2 hr buffer if I'm changing airlines.
Disclaimer: I used to work in reservations at Delta Air Lines.