Source vs Target: where is the weakness?

In a perfect world mating the best source with the best target you can get is the way to go, but what about the imperfect world. Which would be weaker; if mid-fi source was mated with a very high end target where the source was unable to take full advantage of the target OR a very high end source mated with a mid-fi target where the target might get overwhelmed by the source? Which of the "real world" combination would have the chance of sounding better?

Is it better to have more headroom at the target end or the source end?
This should make an interesting thread. There will likely be as many opinions as people.

Although I don't consider myself a Linnie, I do agree with Linn in one respect. You should start upgrading at the source components and work your way forward through to the speakers. This way changes will be cumulative.

Within reason of course.
The source is the key. However, there are real world trade-offs. For example, speakers are harder to sell and ship so should you get great speakers first and hold onto them? Or, what if a really great amp deal came along? Should you then build the rest of the system around it? Since these kinds of situations occur all the time, the pragmatic answer is normally "it depends". That's what makes this hobby so much fun. All else equal, most would argue to start with the source and build it out as Kinsekd suggests (I say "most" based on past threads related to this question). I look forward to other responses.
I agree with Kinsekd - this is the path I followed for my own upgrade and it worked extremely well with very satisfying results. HOWEVER, it requires speakers that are sensistive enough to expose the strengths and weaknesses of upstream components, including cable. This leads us to the all important historical question

Which came first the chicken or the egg?
As long as we're on the subject, "What came first, the chicken or the egg?" Hard to answer! I like "Ozfly's" answer
"it depends." I personally think whole systems should be put together from the get go rather than mixing/matching this and that and hoping things work out. Easilier said than done! Well matched inexpensive components many times out perform and provide greater musical satisfaction than
mega-bucks ill-matched components. Tough question. A great, efficient loudspeaker (without a too low impedance) could pobably made to sound great with mid-fi source/amplification.
P.S. It's critical to have an overall philosophy or approach in mind regardless of how you go. To Kotta's point, the system is the key. For example, do you want to go the SET/efficient speaker route? Are you interested in near field listening? Listen to as many alternatives as possible before embarking on the journey regardless of where you begin.
Isn't the listener the target? Perhaps we should be upgrading ourselves! Easier said than done.
I disagree that someone should start with a source first and work their way to their speakers. Why do I think this? Because ultimately it is one's speakers that determines the limits of what a system can sound like.

You have to find a speaker that you like best first. Next you have to find an amp that matches the speaker. This is a pretty critical match because not all amps can really drive any speaker or sound good driving any speaker.

Next one must find a preamp that matches the amp. Amp/Preamp synergy is amazingly important for good sound. If these two pieces do not work well together, you can forget about good sound.

Last you focus on the source. BY now you have a system that can be transparent enough so you can hear the differences in sources in your system.

If you started with a grat source in a min or low fi setup, you really will have no idea what the final sound is going to sound like.

Jond, you're onto something, but the room is the target. This source v. speakers argument has been done here a number of times. I'm on the speakers first side, with the understanding that it's both a quality (accuracy) and suitability (room) issue. Then source, but everything else is about equal. I could listen to a minisystem driving my favorite speakers, but I could not listen long to the best cd player in the world through cheap speakers.