All objects reflect back whatever light is thrown on them (including light bouncing off of adjacent colored objects). The more strongly colored the object in question, the more it filters this light. Neutral objects (black, gray, white) provide almost no color filtering, that is to say, they absorb or reflect the entire spectrum equally. That means that with a pure white object, what you are seeing is really just the sum of all the light striking it. As a result, the one color that white almost never really appears as is pure white.
But I'm not here to talk about color just yet - as usual, I'm here to talk about VALUE. First and foremost, the reason an element in a painting appears white is because it's the whitest thing in the painting - or the whitest (lightest) thing in a particular part of the painting, where a particular lighting condition has been set up. (This is in fact true of all colors - what appears as blue is simply the bluest thing in the painting, what appears as yellow is simply the yellowest thing, etc., but again, we're not here to talk about color).
Take a look at this painting, The White Bear (featured in Spectrum 18, a full page no less, thank you very much):
There are several "white" elements in it - the woman's head scarf, her apron strings, the bear, the girl's head scarf. Here are those colors sampled, shown on a pure white background:
Apart from the intense orange edge light, most of the painting appears somewhat uniformly lit. At least, all the figures look like they exist in the same space.
But check out how dark the mom and dad figures appear in isolation. The value range for elements from the white apron strings to dad's dark trousers has been extraordinarily compressed - but the apron strings still read as white because they are the whitest thing in that area.
Now with the girl overlaid - it seems like she comes from a different painting:
She'd fit better if mom and dad were painted like this:
Something like the figures in this painting:
Or if the girl were like this, then she'd fit with the darker mom and dad:
So when you're setting up a painting, or struggling to fix one... or struggling with color, or struggling in general, take a breath and first identify the white in each area of your painting. Everything else in that area must fit between it and that areas's black - even if this means you have to deal with incredibly subtle shifts of value. I've said it before and I'll say it again - the illusion of light is about relationships and primarily value relationships, not contrast or saturation.
Be sure to check out Part Two of this series.