Learn English – n English idiom for trying to do two things at the same time and failing at both of them due to splitting your effort

I'm basically searching for the opposite of putting all your eggs in one basket, where the risk is total failure because you did not hedge your efforts. I'm searching for a phrase that encompasses splitting your efforts and evokes the prospect of failing at both. A bird in the hand is worth two in the bush also encompasses a sense of having focused your efforts, but only alludes to total failure.

To satisfy the criteria, the phrase must cover:

• attempting two things (or multiple iterations of one thing)
• both of which you might succeed at, if you addressed them serially
• but both of which you fail at because you address them in parallel

Here are the idioms/phrases that do not sufficiently cover the idea and why:

"A servant to two masters"

This has the sense of competing priorities, but not the sense of failing to achieve your goals.

"Jack of all trades. Master of none."

The sense is insufficiently pejorative. Being a jack-of-all-trades can be seen as an advantage; I'm trying to focus on the lack of mastery.

"The man who fancied himself a wit, and was half-right."

From Christopher Hitchens. Hilarious, but lacking the sense of split effort between two things.

"Walking and chewing gum at the same time."

An idiom for multi-tasking, not for failing to multi-task.

Edit

Addressing a few suggestions and another one that occurred to me:

"Hedging your bets" or "Betting against yourself"

In this case, you cannot win both bets, but you will almost certainly win one. You limit your upside, but also limit your risk. I'm looking in particular for total, catastrophic failure due to split attention/effort.

"Can't walk and chew gum at the same time"

Failing to do two fairly simple things that most everyone can do, not failing at two hard things because you tried to do both.

"Robbing Peter to pay Paul"

In this case, Paul gets paid. If you were to rob Peter to pay Paul, and get robbed yourself before you could get the money to Paul, that'd be closer. 🙂

"Spreading yourself too thin"

An actual English idiom, but missing the implied sense of failure.

"Too many irons in the fire"

"A lot of irons in the fire" can be viewed positively, so this is missing the sense of a strategic misstep.

"Between a rock and a hard place"

The sense is of needing to act but having no good alternative.

"Falling between two stools"

My favorite answer so far because it does appear to be an actual English idiom (not a proverb). In that sense, it's great, but some of the usages are closer to a failure of categorization than a failure of effort. And falling between two stools implies that the attempted action was simply sitting down.

"Chasing two rabbits and catching none."

This is the exact sense I was looking for, but it appears to be a cross-cultural proverb rather than idiomatic English like "putting all your eggs in one basket". I'll be accepting it, and using it in my work! But if there are other ideas, please let me know.