Short answer
Choose the temperature that counteracts the room rather than the one that agrees with it. Rooms with little direct sun, and rooms full of warm materials such as oak, brass or terracotta, generally need a warm neutral. Rooms with strong direct sun, or with cool materials such as grey stone, chrome and marble, can take a cool neutral without feeling clinical.
A neutral is never colourless — every white, grey and beige contains a small amount of pigment that gives it a warm or cool bias, and because the bias is subtle it is amplified rather than hidden by the room around it. A cool neutral in a north-facing room adds cool to cool and reads as grey and unwelcoming; a warm neutral in a bright south-facing room adds warm to warm and can read as yellow or dirty. Counteracting the room is what produces a neutral that actually reads as neutral.
Which way does it face, and what large materials are in it? A north-facing room with oak flooring is a warm-white room; a south-facing room with grey porcelain tile is a cool-white room.
Printer paper is close to a true neutral white. Against it, a warm neutral shows yellow or cream and a cool one shows grey or blue. This is the fastest way to see a bias that is invisible in isolation.
Neutrals of different temperatures side by side is the most common near-neutral mistake. If the walls are a warm neutral, the ceiling and trim generally need to be the same colour or a lighter version of it — not a brilliant white from a different range.
Sanitaryware, appliances, uPVC window frames and radiators are usually a cool, bright white and cannot be adjusted. A warm white beside them will read as noticeably yellow.
Use brilliant white with care anywhere other than a ceiling. It contains optical brighteners in many formulations and can read as harsh against almost any natural material — it is a specification choice for maximum light reflectance, not a default.
How to test it
Test neutrals in pairs, never singly, and always against a sheet of plain printer paper as a reference. Include the ceiling and trim colour in the same test — the relationship between them is the thing that will actually be visible.
Undertone
Neutrals carry undertones beyond simple warm and cool: greens, pinks and violets are all used to soften whites and greys, and these become obvious at scale even when invisible on a card. This is why testing a neutral large matters more than testing a strong colour large, not less.
Lighting
Artificial light has a colour temperature of its own, and it interacts with the paint's bias. A warm neutral under a warm bulb can read as distinctly yellow after dark even when it looked correct in daylight.
Finish
Near-neutrals show sheen more than any other colour, because there is no depth of colour to distract from the reflection. A satin or eggshell white on a wall with any unevenness will show every imperfection under raking light.
Whatever a calculation or a chart says, view a physical sample in the actual location before committing to a large surface.
Colourwise editorial guidance, drawn from conventional decorating practice and from the physical behaviour of light and surfaces described in the linked lighting and finish guides. Not manufacturer advice — always follow the specific instructions supplied with any product you use.