I've been making chicken broth pretty much the same way for decades. I cut up a whole chicken, cutting through the bones of all but the breast and thighs (to release marrow). I pretty deeply brown the pieces (with skin) in the bottom of a stock pot, then remove. I then saute roughly chopped onion, celery and carrot in the chicken fat until browned, then I deglaze with a little wine or sherry. The chicken pieces go back in the pot with a gallon of water, I bring just to a boil, cover and turn off the heat. Now it sits for a half an hour or so. I then fish out the breasts, refrigerate the breast meat, and return the bones and the skin to the pot. I simmer the remaining pieces for about another 40 minutes, remove the thighs, reserve the thigh meat, break the bones, return the skin and bones to the pot and slow simmer everything for the better part of the rest of the day. I allow the broth to cool, strain through a colander then a fine sieve, then refrigerate overnight.
Here's what's weird. Half the time, I can simply remove hard solid fat from the top of the chilled broth. The other half of the time the fat separates and thickens, but never solidifies, I end up dirtying my fat separator. I can't figure what could possibly be different from one batch to the next. I generally even buy the same brand of chicken and I only add a tiny bit of oil (vegetable) for the initial browning.
Even weirder, sometimes the broth is so full of gelatin that it is actually harder than the fat, it makes for a good soup when that happens, but makes separating the fat a bit of a pain.
Best Answer
It's down to what the chicken ate while it was alive. Saturated fat sets, olive oil sets if you chill it, but not otherwise, and a number of seed oils do not set (rapeseed for instance). When you make a stock which has solidified fat on top, that's saturated fat, so I'd hazard a guess that the stock where the fat doesn't set means a healthier eating chicken, because it contained less saturated fat to start with.
UPDATE: Thank you to the person who bothered to do the research and said my answer 'might have some merit'. Chickens are no different from human beings - the fats you put in are the fats floating round your bloodstream and depositing in various places; think about corn fed chickens, where the fat composition is slightly different, not to mention the colour of the flesh itself. That will be a partial explanation; when taken together with the fact that not all chickens, even in the same flock or brood, get to eat the same diet, because the pecking order dictates that some free range birds don't always get the pick of the food, explains differences in chickens from the same supplier. Of course, if you can come up with another explanation, I'd be delighted to hear it...
UPDATE 2: Perhaps I should have been clearer. I am not for a moment suggesting that the fats eaten are deposited in their original form, but if you know anything about biology (chickens or otherwise) then you'll know that certain synergies occur, depending what's put in, which change the composition of any fats deposited within the body system. Hence the connection between eating lots of saturated fat and having high cholesterol in humans, for example.
UPDATE 3: Rumtscho: Can't find any scientific evidence so far to prove this theory regarding chickens, but, for interest's sake, and to prove how much of a difference it can make, farmed salmon in Britain no longer has a balanced omega 3/6/9 ratio, as it should do, and still does in the wild. It's because the feed had to be changed, and the consequence of that has been a much higher level of omega 6 in particular. I'm still looking for something on chicken.
UPDATE 4: Now I've had time to look properly, it's not at all difficult to find scientific evidence, there's plenty of it. There's a study carried out by The American Society for Nutritional Science in 2000 comparing the fat deposition (and other metabolic processes) between chickens fed the same diet, but one lot with saturated fat included in the form of tallow, and the other lot with polyunsaturated fats. The fat deposition in the birds fed tallow was greater, and the composition of the fat contained more saturates compared to the polyunsaturated group. These results reflect previous studies (Sanz et al 1999 and 2000).
Effectively, it's like everything else - you get back what you put in.