I believe the definition section of the following wikipedia page will be of use:
Your teacher is correct to say Hooke's law is only really valid up to the proportionality limit, and this is simply because Hooke's law is a proportionality law, i.e. Force (or stress) is proportional to extension (or strain). Beyond the proportionality limit, by definition the force and extension are no longer proportional and so Hooke's law is not entirely valid.
However, the elastic behaviour itself, where deformations are temporary, lasts up to the elastic limit. This does indeed mean there is a region where you obtain non-proportional elastic behaviour, between the proportionality and elastic limits. This non-linearity arises here because, in my best understanding, Hooke's law is simply an approximation that is accurate until you approach the limit of elasticity. For metals, like steel, this non-linear elastic region is usually small, and in some practical cases it is ignored altogether! However, materials like elastomers will have a much larger non-linear elastic limit.