During both the Principate and Late Antiquity, soldiers who were unable to perform military duties because of wounds or disease were invalided out of the army. They received a form of honorable discharge, as well as a “severance package”, though these benefits were usually less than if they had served their full term of service.

However, based on the phrasing of your question, it seems like you’re more interested in the experience of soldiers who, despite being in perfect health, couldn’t quite handle the army’s demanding physical environment, a bit like Private Pyle from Full Metal Jacket. To be quite honest, I have not seen any reference, either in ancient or modern references, to recruits "washing out" of military training. While I invite other experts to comment (paging /u/Celebreth), it is likely that instances of washing out did not occur with any regularity. Desertions certainly did occur (and scholars debate whether or not the levels of desertion increased during Late Antiquity), yet it is important to distinguish between men who actively sought to leave military service (desertion) and those whom the army deemed not up to standards (washing out).

Most recruits, both during the Principate and Late Antiquity, came from rural, agricultural backgrounds. Although recruits certainly were drawn from cities, by and large, the majority of recruits were rural, coming from poorer and more rural areas of the Empire. These men were already conditioned to lives of grueling physical exertion (pre-modern agriculture being extremely labor-intensive). Military life and training was thus not that much different, in terms of levels of physical activity, than their pre-military lives. Indeed, with it’s regular meals, excellent medical care, and pay (even if low), military service offered many attractions. Additionally, the army often engaged in “targeted recruitment” (for a variety of reasons) from non-Roman peoples (“barbarians”). These groups were targeted in part because of strong martial cultures deemed conducive to producing quality recruits. These recruits were even less susceptible to handling army life, especially when considering they also came exclusively from rural backgrounds. With this in mind, it is highly unlikely that recruits would have found the army’s physical standards too demanding. While certainly intense, military life and training was not that much more demanding than rural, agricultural life, and offered many tangible benefits. It is therefore unlikely that recruits were unable to handle military life.