An upper bound, at least.
First, forget terms of whole bottles for a minute. A "bottle" is actually a weirdly-named, infinitely-divisible item that can be consumed and then traded.
Each bottle of cola trades for $1/3$ of a bottle of sprite. But wait, if each bottle of sprite is actually worth $1/4$ of a bottle of cola, you have to figure that in, too! So really, given enough time, each full bottle of cola is worth $1 + 1/3 + 1/12 + 1/36 + 1/144 + \cdots$ empty bottles. Some re-arranging yields $(1 + 1/3) * (1 + 1/12 + 1/144 + \cdots)$. Similarly, each bottle of Sprite amounts to $(1 + 1/4) * (1 + 1/12 + 1/144 + ...)$.
If you have $N$ of each kind of bottle, the grand total is $N * (2 + 7/12) * (1 + 1/12 + 1/144 + \cdots)$ or $31N * (1/12 + 1/144 + \cdots)$ bottles. As it turns out, $(1/12 + 1/144 + \cdots)$ is actually $0.11111\dots$ in base 12, which (like with base 10 and $1/9$) works out nicely to $1/11$ in decimal.
So, ultimately. the upper bound given $N$ of each type of bottle is $31N/11$.
That's quite a neat number, but it was computed with fractional bottles in mind. Fortunately, in the real world, we can borrow! Having some large number of one kind of bottle allows you to borrow up to $1/12$ of that quantity, trade back and forth, and then repay. I suspect (but am not totallty sure) that trading allows us to get up to that upper bound rounded down. In the case of $N=120$, that means we can drink $\lfloor338.18181818\rfloor = 338$ bottles.