Can I fly via a third city on a second ticket?

I do this all the time and I always book the C-B flight connecting to the next A-C flight. I am rarely time constrained and spending a day in a hotel is not a big issue for me (I work remotely anyways). Yes, this means the cost of the hotel needs to be factored in. However, as mentioned in the original post, you might be doing this for other reasons than finding the cheapest possible flight and I often do. Although my examples are also for cost reasons but nowhere near the cheapest economy tickets.

I was flying from Vancouver to Wellington so I took Westjet to Honolulu, slept there, then used Jetstar to Wellington via Sydney. As I really hate economy seats for longer flights, on Westjet, I had an extra legroom seat (doable for six hours) and on Jetstar to Sydney I was flying business class (only thing I would do for such a long flight). Buying a single business class ticket from Vancouver to Wellington was about three times as much as what I spent on this. Not to mention that flying from the west coast to Australia/New Zealand is 15-ish hours and I can’t imagine doing such a long flight: I’m OK w/ 10 and get antsy on 12 hours even in business. 6, sleep, 10.5 is much better for me.

Another example. Delta had a crazy business seat sale for Seattle-Frankfurt and my destination was Budapest so I took the bus from Vancouver to SeaTac then slept there, took Delta to Frankfurt, slept there (see a pattern?) and then took Lufthansa to Budapest the next day.

Yes you can do this but be aware of the following things:

1. You might have completely different (harder-to-obtain) visa requirements, check carefully before!
2. In case of a delay of your first flight (A to C) you loose your second ticket (C to B) under most ticket conditions.
3. If you travel with check-in luggage you likely have to re-check your bags. This more often than not involves going through immigration which involves issues mentioned above in 1.

In more detail:

1. If travelling on one ticket (i.e. A to B) chances are you will only need a transit visa if at all (but this is in no way certain, so check! This guy was lucky since he already had a US visa, otherwise he would have needed Schengen short stay). On two different tickets, if you have to pass through immigration to retrieve and re-check your bags or if a part of the A to C itinerary is domestic / intra-Schengen, chances are you need a short-stay or similar visa. If you want to know whether you need a visa, see here for Schengen, here for USA, here for UK or here for Dubai just to name a few of the usual suspects.
2. If you buy a ticket from A to B (on a single ticket) then the airline that issues the ticket commits to getting you to B in the end, even in the case of delays/ missed connections/ etc.
   If however you buy two separate tickets, your first flight is late so that you miss your second flight, the airline that issued the second ticket can (I can only cite again from this excellent answer):

then claim (correctly) that you broke the terms of the CoC on their ticket by not arriving at the airport in time, and thus would normally cancel your ticket. Depending on the airline, they may choose to show you charity and accommodate you on a later flight (eg, next day), but they have no responsibility to do so, and if the next days flight is full, you’re pretty much out of luck! In this case most (but not all!) travel insurance policies would cover you, although some may only cover you to the price you originally paid for the missed flight, even though the last-minute replacement flight could cost significantly more – so read the fine print on the policy!

2. (continued). Finally let me add that this might be another reason for visa problems if due to a delay you end up stranded at C.
3. If you have to re-check your luggage, many airports do not allow you to retrieve or check-in luggage from the international transit area, so if you need to re-check your bags don’t only count the time for doing so (possible further delays) but also having to go through immigration (possibly twice: in & out, so see 1. above). If your bags are seriously delayed your carrier of the first ticket might even not be able/willing to ship your luggage all the way to C where you want it.
   Some airlines will have interlining agreements and in that case you can ask at A to have your luggage checked in all the way to B through C, showing them your second booking. I have no idea what is the success rate of this as in some cases they might refuse / be technically unable to do so, but it does not hurt asking.

Finally, you should add up the hidden costs (visa, hotel stays, cost for alternative travel in case your plan fails) and compare that to the A to B ticket price.

On the other hand there’s nothing wrong with getting two different tickets if you know your risk. You might have alternative options in case or you wanted to stay a night or two in C anyway. Happy travels!