If X=Y+Z, Is it ever useful to regress X on Y? If we have X and Y that are mathematically dependent: X = Y + Z, is it 'forbidden' to use Y as a predictor to X in linear regression?  I'm trying to find a concise explanation for why it is, or isn't.
One explanation I've found is that you shouldn't use simple linear regression in this case, but multiple linear regression, with Y and Z as predictors, but this doesn't seem right. The mathematical relationship is exact, and the multiple regression would return coefficients 0, 1, and 1  for X = 0 + 1Y + 1Z. So, there is something more fundamental I'm missing.
The purpose for linear regression is predicting the value of one variable from the other. But we already know that X can be calculated as the sum of Y and Z, so effectively, we are calculating the regression between Y and Y + Z. Is this 'forbidden'?
But, if we can only measure Y, does the regression give us a tool to predict X? Isn't it better to find the regression between Y and Z? I'm confused.

Linear regression is a tool that is used to achieve a goal. So any answer will depend on the goal to be achieved. As said already in the answer of @Noah, if you already know $X$, $Y$, and $Z$, I can't see any goal worth achieving that is achieved by this regression. Why would you want imprecise predictions of $X$ if you can have precise ones?
If however you don't know $Z$, linear regression may work well for predicting $X$ from $Y$. There is nothing that formally forbids you from trying that out (in fact if $Z$ is independent of $Y$ and distributed according to standard regression model assumptions, the standard regression model is just fulfilled). But then, depending on the exact nature of the data (particularly the distribution of $Z$ and how it is related to $Y$), it may not work that well, and/or other techniques may work better. This can be explored for example using bootstrap or cross-validation.
