Is this hopeless?  

"Sanding is alway the LAST...
"Sanding is alway the LAST thing to do. Only if the other options didn't give good results."
Dear Patrick and Alix, as professional you are able to see at first view how a certain damage has to be treated, then it's not necessary at all to adopt ridiculous measures as described above.

"as professional you are...
"as professional you are able to see at first view how a certain damage has to be treated"
We can agree with that but only if you see the piece in real. It's highly hazardous to make that kind of statement based only on a tiny picture. In general and especially for inexperimented people, it's safer to recommend the soft, graduated approach.

Patrick, you can be sure...
Patrick, you can be sure I've seen and handled that kind of damage already more than a hundred times.
You are right it's better to see the piece in real before to issue a statement, exactly this fact should make the people wonder who aren't professional.
Just read the stuff above, it's unbelievable.

Re: Acerbic Wit
The fence/field bit is a fine snippet of creative writing.
"Snap", so to speak-

well, I
have had similar results to TCHP. Really quite noteworthy.
If you have a veneered piece w/ water damage/marks, the Barkeeps friend is, in fact, your friend. Yes, you may need to do some finish work, but is that really all that surprising?
To each their own, of course, but I cannot see how anyone would consider this method as not an option.

.
My apologies to pubpat. Yes you saw what everyone else missed - but as 99% of us are idiots its no surprise - that the answer is to start sanding and don't stop til you see daylight. Seem so simple now I think about it. The whole antiques restoration business must be worried now the lid is off such a closely guarded secret of their profession. Well we all know where to head when we need restoration advice in future. In fact I'll go further and be the first to propose that when the new forum is up and running there should be a special 'Ask Uncle Pat' thread where we can all bring those troublesome renovation woes to you. Maybe you could even have your picture above it - just you with an Eames chair, an angle grinder and a blow torch.

I must admit it was my mistak...
I must admit it was my mistake that I didn't follow the instructions correctly. I remember reading paulanna's earlier comment about scrubbing with naptha with a toothbrush and somehow the idea of scrubbing stayed in my head when I decided to try tchp's approach due to his amazing results. That's how I ended up blending the 2 methods together and created an inferior outcome. The picture shows the slight fading of the finish near the edge of the top.
One thing I truly believe is everyone here's trying to help and I"m sure there's more than 1 way to success. I think I mentioned before that I suck at DIY'ing but just love to learn. Hopefully everyone will continue to contribute, preferably with less tension.
Thank you all again.

If the stain truly is a...
If the stain truly is a mildew or some other organic stain, you should still try tktoo's bleach advice, trying a small amount on a q-tip.

And my sincere apologies to all.
When I suggested that we should all perhaps be a bit more careful, I meant in both offering and accepting advice. I certainly did not mean to disparage anyone for trying to help or otherwise precipitate a row.

Work in progress
I spent a couple hours on the seat panel yesterday. I started with bleach mixed with water, but it didn't do anything to the stains after 3 tries. So I went ahead with BKF again without scrubbing. I mixed it with water and made a thick paste and tested it on a small area. When I saw no ill effect, I spread it on top of the panel for a minute or 2 the first couple runs. Then I'd wipe it clean with a damp cloth.
I repeated that process about 7-8 times. After each try, more stain would come off. The picture was taken this morning after the wood has dried. It looks similar to when I scrubbed the back of the top panel with the finish coming off.
Here're a couple of thoughts:
1: the faded finish is a result of the stain, but not scrubbing because the same thing happened even when I didn't scrub.
2: the faded finish can be reversed. I used "Howard Restore a Finish" and it turned out very well. I tried it on the back of the top panel and will do the same with the seat later on.
So far, I'm pretty happy with the progress and will show you the end product when I'm there. Thanks again for all your support and guidance.

Looks like an improvement, without losing
the original surface!

!!!
Most excellent, indeed!~Great thread~Happy ending.
Aunt Mark
(edited to correct my poor spelling...sorry).