The slogan adopted by my old school had adopted was the Latin phrase nihil labore difficile. They claimed that it meant that "nothing is difficult with hard work". However, is this slogan grammatically correct? Shouldn't it read nihil labore cum difficile?
The slogan nihil labore difficile is grammatically correct, but ambiguous and unclear. But being ambiguous and unclear is not at all unusual for a motto.
If we read labore as an instrumental ablative, the slogan means "nothing is difficult with work", as intended. I will return to this phrase in a moment.
If we read labore as an ablative of respect, the slogan means "nothing is difficult concerning work", which I would interpret essentially as "all work is easy". The Latin ablative is very versatile, and especially in a compact motto the exact meaning is open to interpretation. Perhaps there are more readings than these two, but they at least illustrate the ambiguity.
If one adds the preposition cum to the ablative labore, the instrumental reading is still natural but the other one is ruled out. The form nihil cum labore difficile would resolve the ambiguity, but ambiguous is not the same as wrong. A motto often prefers brevity at the cost of clarity.
The phrase "nothing is difficult with hard work" is logically a little weird: it is not unusual that "difficult" means "requiring hard work". But it can also be read charitably as "anything is possible if you work", and logical issues do not always matter for translation. The message is not very clear, but with the help of context and good will the slogan can be understood to state that working hard will allow you to reach what you want.
The claimed meaning does make sense to me, but it could have been put more clearly in Latin and the Latin phrase does leave room for other interpretations.