Household bleaches that you buy from your supermarket may seem the solution to whiter-than-white whites and some of us will have been using them for so long that we no longer give it a thought but in fact, over time, bleaching corrodes clothing and even reduces the clothings' natural whiteness; whites develop a dull yellow hue quicker than they would had you not used bleach at all.
Even household detergents contain a fluorescing agent to make whites seem whiter (known as ’optical brightening’) and is one of the culprits if you are sensitive to some detergents. The effects of flourescing are short-lived - you need to keep using it to maintain the bright-white effect, whilst natural whiteners last, well... naturally. Hence, white cotton or linen washed in household detergent and stored for a lengthy period, which appears dull or not quite so bright when removed from storage, has probably simply lost its fluorescent hue.
The following natural whiteners are listed in safe-use order, from the most innocuous to those which must be used sensibly and with caution (just as bleach must be). Crucially, each also advises what they must NOT be mixed with.