The only way I can make sense of the question after the exchanges in the comments is that you're asking whether there exist two quantities such that their sum is finite but the quantities themselves are not finite. The answer is no.
There are basically two ways of regarding limits. In one, a limit is said to exist only if it exists in the strict sense that there is a finite number to which the sequence converges. Under this paradigm, already saying that the sum of the limits is finite implies that the limits exist and are thus by definition finite.
In another way of speaking, a limit can be said to exist and be $\infty$ or $-\infty$ if the sequence diverges to $\infty$ or $-\infty$, respectively. Under this paradigm, limits are no longer real numbers and cannot necessarily be added. While addition can be extended by defining $\infty+x=\infty$ and $-\infty+x=-\infty$ for any real number $x$, it cannot consistently be extended to define $-\infty+\infty$. Thus, even under this paradigm, saying that the sum of the limits is finite implies that the limits are finite, since the extended addition operation only yields finite numbers for finite operands.