I voted "No" before really thinking about the question. I came here from your other poll "Do earthworms feel pain and are therefore of moral concern?", which I'm assuming inspired this poll. Upon further consideration, I would probably change my answer to "Other", but I want to provide some clarification:
If you simply take "Suffering" to mean "the experience of pain", then of course suffering necessarily accompanies pain, because they are one and the same (though I think it would be wrong to say the pain causes the suffering, if this is what you mean, as things do not cause themselves). I think this is reasonable definition of suffering, as this is how it is used colloquially.
However, in ethics, we often use "suffering" to denote the opposite of "flourishing", i.e. what makes life go poorly rather than what makes life go well. A life full of suffering would be a bad life, just as a life full of flourishing would be a good life. If this is the sense of "suffering" that we want to use, then it is not true that pain always causes suffering, because there are instances of pain which do not make your life go any worse, e.g. when you regard the pain wholly favourably, such as a masochist who loves being slapped.
If one says, "Alright, well, I am using the first definition of suffering, in which it just means the experience of pain," that's fine, but I think it's important to note that it doesn't answer the earthworm question, because by making suffering synonymous with the experience of pain, we have not yet answered the question of whether pain is always a moral bad, and it seems plausible that it is not (as in the case of the masochist).