"Short of" generally conveys the idea of lacking or not having enough of something, while "short on" often implies a temporary shortage or insufficiency. However, these distinctions can be subtle, and in many cases, both prepositions can be used interchangeably without causing confusion or a significant change in meaning.

For example;

I'm short of cash/money.

I'm short on cash/money.

Both sentences convey the idea that you do not have enough money at the moment. While the first sentence with "of" might imply a more severe or lasting lack of money, the difference is subtle, and both expressions are commonly used.

Regarding your examples;

I'm short of breath

Here, using "of" is correct. Using "on" would indeed be incorrect in this context, because it doesn't make sense to be temporarily out of breath.

We are short of/on good players in our team

Both "of" and "on" can be used here, with "of" possibly implying a more persistent lack of good players, while "on" could suggest a more temporary shortage.

The hospital is short of/on nurses

Again, both prepositions are acceptable, with "of" potentially conveying a more lasting or systemic issue, and "on" suggesting a more temporary or circumstantial shortage.

In conclusion, there are indeed some nuances when using "of" and "on" in these contexts. While they can often be used interchangeably, the preposition "of" might suggest a more persistent or severe lack of something, while "on" could imply a temporary or circumstantial insufficiency. However, it is important to note that these differences are subtle, and in many cases, using one preposition over the other will not significantly change the meaning of the sentence.

"Short of" generally conveys the idea of lacking or not having enough of something, while "short on" often implies a temporary shortage or insufficiency. These distinctions can be subtle, and in many cases, both prepositions can be used interchangeably without causing confusion or a significant change in meaning.

For example;

I'm short of cash/money.

I'm short on cash/money.

Both sentences convey the idea that you do not have enough money at the moment. While the first sentence with "of" might imply a more severe or lasting lack of money, the difference is subtle, and both expressions are commonly used.

Regarding your examples;

I'm short of breath

Here, using "of" is correct. Using "on" would indeed be incorrect in this context, because it doesn't make sense to be temporarily out of breath.

We are short of/on good players in our team

Both "of" and "on" can be used here, with "of" possibly implying a more persistent lack of good players, while "on" could suggest a more temporary shortage.

The hospital is short of/on nurses

Again, both prepositions are acceptable, with "of" potentially conveying a more lasting or systemic issue, and "on" suggesting a more temporary or circumstantial shortage.

In conclusion, there are indeed some nuances when using "of" and "on" in these contexts. While they can often be used interchangeably, the preposition "of" might suggest a more persistent or severe lack of something, while "on" could imply a temporary or circumstantial insufficiency. It is important to note that these differences are subtle, and in many cases, using one preposition over the other will not significantly change the meaning of the sentence.

"Short of" generally conveys the idea of lacking or not having enough of something, while "short on" often implies a temporary shortage or insufficiency. However, these distinctions can be subtle, and in many cases, both prepositions can be used interchangeably without causing confusion or a significant change in meaning.

For example;

I'm short of cash/money.

I'm short on cash/money.

Both sentences convey the idea that you do not have enough money at the moment. While the first sentence with "of" might imply a more severe or lasting lack of money, the difference is subtle, and both expressions are commonly used.

Regarding your examples;

I'm short of breath

Here, using "of" is correct and more common. Using "on" would indeed be incorrect in this context, because it doesn't make sense to be temporarily out of breath.

We are short of/on good players in our team

Both "of" and "on" can be used here, with "of" possibly implying a more persistent lack of good players, while "on" could suggest a more temporary shortage.

The hospital is short of/on nurses

Again, both prepositions are acceptable, with "of" potentially conveying a more lasting or systemic issue, and "on" suggesting a more temporary or circumstantial shortage.

In conclusion, there are indeed some nuances when using "of" and "on" in these contexts. While they can often be used interchangeably, the preposition "of" might suggest a more persistent or severe lack of something, while "on" could imply a temporary or circumstantial insufficiency. However, it is important to note that these differences are subtle, and in many cases, using one preposition over the other will not significantly change the meaning of the sentence.

"Short of" generally conveys the idea of lacking or not having enough of something, while "short on" often implies a temporary shortage or insufficiency. However, these distinctions can be subtle, and in many cases, both prepositions can be used interchangeably without causing confusion or a significant change in meaning.

For example;

I'm short of cash/money.

I'm short on cash/money.

Both sentences convey the idea that you do not have enough money at the moment. While the first sentence with "of" might imply a more severe or lasting lack of money, the difference is subtle, and both expressions are commonly used.

Regarding your examples;

I'm short of breath

Here, using "of" is correct. Using "on" would indeed be incorrect in this context, because it doesn't make sense to be temporarily out of breath.

We are short of/on good players in our team

Both "of" and "on" can be used here, with "of" possibly implying a more persistent lack of good players, while "on" could suggest a more temporary shortage.

The hospital is short of/on nurses

Again, both prepositions are acceptable, with "of" potentially conveying a more lasting or systemic issue, and "on" suggesting a more temporary or circumstantial shortage.

In conclusion, there are indeed some nuances when using "of" and "on" in these contexts. While they can often be used interchangeably, the preposition "of" might suggest a more persistent or severe lack of something, while "on" could imply a temporary or circumstantial insufficiency. However, it is important to note that these differences are subtle, and in many cases, using one preposition over the other will not significantly change the meaning of the sentence.