It can certainly be less expensive to purchase separate tickets for certain segments as opposed to the complete journey, particularly on a mixed itinerary.
I would strongly advise against this unless you are a very experienced international traveler, for a number of reasons:
Checked baggage - Even if the two airlines you are flying have an interlining agreement, the agent may still refuse to check them through if the segments are on separate tickets. Thus, you would need to claim your baggage, get processed by customs and immigration, and then stand in line to have them re-checked to your next destination.
Flight irregularities - Since the two tickets are independent of one another, agents from the first airline will be unable to protect you on your connecting flight in the event of weather, mechanical problems, air traffic conditions, labor actions, or any of the other things that can cause your first flight to be delayed or canceled.
Indeed, airlines adjust their schedules constantly, and you may find that your first flight has been shifted three hours later, making the connection impossible; you would be responsible for contacting the second airline to make arrangements and to pay any associated change fees. This is of particular concern on international flights, where many routes have frequencies of once a day or even less.
If the flights are on the same record (PNR), you avoid the above scenarios, as the agent would have no reason to refuse interlining your bags, and in the event of problems with any segment that causes you to misconnect, the airline you purchased with would be responsible for booking you on alternative flights. Otherwise, in a worst case scenario, you could be forced to pay thousands of dollars for a walk-up fare to continue your journey. Given that, a few hundred dollars seems like cheap insurance.
I will say that these risks are not serious when booking your outbound and return trips on separate tickets, unless you have a very short turnaround time. It is not necessarily cheaper to do it this way, and you may lose some benefits. For example, a United Airlines round trip award ticket allows both a stopover and an open jaw, but not on two one-way awards, even though each requires the same number of miles.