There have been massive increases in the supply of prescription and nonprescription opioids, the prevalence of opioid use disorder, and rates of fatal and nonfatal unintentional poisonings or overdoses in the US. We examined the relationship between rates of unintentional overdoses and intentional overdoses (poisoning suicides), using data for the period 2005-16 from the Centers for Disease Control and Prevention. Contrary to expectations, we found no evidence of positive associations in their trends. While unintentional opioid overdoses have increased dramatically, rates of poisoning suicides have scarcely changed. Furthermore, while unintentional overdoses have increased the most among younger males, poisoning suicides have risen the most among older females. We found that the prevalence of opioids in poisoning suicides was high but did not change notably, nor did we find the large shift to heroin or fentanyl that has occurred in unintentional poisonings. There is growing interest in the potential links between suicide and opioid overdose deaths, yet these results suggest that the relationship between them is not straightforward.