In the figure above you see the relation between the number of half sib offspring with observations for estimating the breeding value of their parent, and the accuracy of the EBV for that parent. The lines are in order of the heritabilities. For a trait with high heritability (0.8) 10 offspring is sufficient for an accuracy of 0.85, whereas you would need 48 offspring to achieve that for a trait with heritability of 0.2. The more information is available, in this example many offspring, the higher the accuracy of the EBV. Even with low heritability, the accuracy eventually will approach 1. Though for low heritabilities you would need very (often unrealistically) large numbers of offspring: with a heritability of 0.2 and 100 offspring the accuracy is ‘only’ 91.7%, and with 200 offspring it only increases to 95.6%. A general rule is that the higher the accuracy of the EBV is, the less likely it is to change if additional information (more offspring) becomes available. Traits related to fertility often have a low heritability. For those traits with mass selection the accuracy of the EBV will be low. However, if very large numbers of offspring can be produced, such as in dairy cattle (bulls), pigs (boars), or poultry (hens and cocks), the EBV eventually can be estimated very accurately.
Thus: even for traits with low heritability the accuracy can increase to 1 if information on sufficiently large number of offspring is available.