The most important thing for ensuring are comfortable on the bicycle is making sure your bicycle fits you and is set up for you. If your bicycle is set up badly, your saddle will feel uncomfortable, your knees will hurt, and your neck and shoulders will hurt ... and you will be in pain.
I used to ask new cyclists if their bicycles fit them, and they would come back and tell me that someone at their LBS told them the bicycle fit, and so I'd leave it at that. But this year I've started digging further and have discovered what people mean when they say that.
What they mean is that they went in to the shop to buy a bicycle, they stood over several bicycles, the guy at the LBS said, "Yep, that should be about right." ... and they rode off with their new bicycles, without ever having their bicycles set up for them. That might be a good starting place, but that's not all there is to fit. Then these same people come to BF and complain that they hurt when they ride.
First, make sure your saddle is in the right position. Make sure it is high enough, without being too high. Make sure it is far enough forward or backward. Make sure your handlebars are in the right position. Your body and the bicycle are a machine ... if you've ever studied subjects like physics, strength of materials, motion, and so on you will have discovered that there is an optimum set up for machines so that they use the least energy to put out the most energy ..... and that's the set up you're shooting for with you on your bicycle.
If your LBS won't do it, there are a number of sites online where you can get a pretty good idea of how to set up your bicycle ... and there's always tweaking involved. Also, the setup seems to evolve. You'll get it just right one year, and then you'll do something like burn your foot to the bone and have to go through months of not walking or moving, and months of physio to learn to walk again ... and when you get back on your bicycle again, it doesn't fit quite right. Or maybe you gain or lose weight so you hold yourself differently on the bicycle. Things do change, but if you have managed to get it set up just right, any changes are usually just minor tweaks.
Once you've got it just right ... take measurements and record them somewhere. Note that most measurements are taken to the centreline, not to the edge. So if you're measuring the distance from the top of your saddle to your bottom bracket, you'll measure from the top of your saddle to the middle of your bottom bracket. It's an engineering/drafting thing.
This is a page of a collection of articles which might be useful to you as you work on getting your bicycle set up correctly. And back when I was starting, the library had resources.
http://www.cyclemetrics.com/Pages/Fi..._fit_links.htm
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And in answer to this question:
"Can a bike with all the available adjustments not be made to fit a person, is that really possible?" --- Yes, absolutely.
Bicycles come in different sizes. When I first started cycling "seriously" I was on a 58 cm bicycle. I set that bicycle up as best I could using the books and various resources available to me and it wasn't bad ... I was able to ride it with minimal pain for shorter rides (100 miles and less). But the bicycle was way too big for me, and I never felt really comfortable on that bicycle.
When I finally had a little bit of money at my disposal, I bought a bicycle ... a 48 cm Cannondale. I knew that the 58 cm bicycle was too big for me, and I thought maybe the 48 cm bicycle would be better. But it was too small for me.
Here's me on that bicycle ... you can obviously see I made adjustments to try to get that bicycle as close as possible to fitting me, but that it didn't really fit at all.
Next I got a 50 cm Giant and that fit quite well, but I still felt just slightly uncomfortable. I had to make a number of adjustments to make it fit better, and I've got it set up fairly well now.
Finally I went custom and I'm on a 49.5 cm Marinoni. Even with that bicycle I had to customize it to get it just right. But that bicycle fits me better than any bicycle I've ever had.