The use of non-orthogonal signals has several benefits over orthogonal signals in multi-coded communications. We provide a novel, theoretical study of non-orthogonal signaling to expand the applicability of these schemes. Motivated by a class of multi-carrier spread spectrum systems, this paper presents a thorough symbol error rate analysis of the broad class of multi-code signaling methods when they make use of codes which are not necessarily orthogonal. Our analysis is also extended to the case where the code set includes the negative of each code vector, i.e., an extension to biorthogonal signaling. Moreover, it is shown that the symbol error rate results derived in this paper reduce to those available in the literature when the multi-codes are orthogonal or have equal correlation between vectors. Additionally, we show how Monte Carlo integration can be used to evaluate the integrals in the error probability calculation and derive low complexity upper bounds on the error probabilities. We show that by combining these techniques, the error probability can be efficiently computed across the full SNR regime. Finally, we use the upper bound of the error probability to develop some analytical insights about the impacts of non-orthogonality among the code vectors on the symbol error probability.