Polysubstance use of alcohol, cannabis, and nicotine has been shown to be correlated with opioid use disorder (OUD). The goal of this study was to determine whether alcohol use disorder (AUD), cannabis use disorder (CUD), and/or nicotine dependence were associated with concurrent OUD. Data came from the 2015–2019 National Survey on Drug Use and Health (n = 282,768, 48.5% male). Weighted logistic regression was performed for experiencing OUD in the past year concurrent with AUD, CUD, nicotine dependence, all pairwise interactions, the three-way interaction, and demographic covariates. Compared to individuals with no substance use disorder (SUD), individuals with AUD had 5.24 times the odds (95% CI [4.25, 6.46]), individuals with CUD had 6.69 times the odds (95% CI [5.13, 8.72]), and individuals with nicotine dependence had 7.12 times the odds of experiencing OUD (95% CI [6.10, 8.32]). Individuals with either AUD and CUD or AUD and nicotine dependence had approximately 15 times the odds of having OUD than those with no SUD (95% CI [12.58, 19.53] and 95% CI [11.63, 18.19], respectively). Individuals with CUD and nicotine dependence had 27.35 times the odds of having OUD than those with no SUD (95% CI [21.88, 34.19]). Individuals with AUD, CUD, and nicotine dependence had 47.31 times the odds of having OUD compared to individuals with no SUD (95% CI [36.79, 60.83]). A multiplicative effect was present when two or more SUD occurred simultaneously and was greatest when all three occurred at once, which suggests that prevention programs or interventions aimed at OUD should focus on persons with multiple SUDs.