Actually, there are scenarios where something might not be considered accurate, but still would not be considered false.
Consider when there is no claim to absolute accuracy. If not, you could not really call the statement "false", although you could call it factually "inaccurate" or scientifically "inaccurate" or historically "inaccurate", etc. For example, if there is a poem which is using figurative language. If it happens to get something scientifically wrong in its metaphors or symbolic language, you would not say the poem was "false" or even "not true" even though you could point out that it might be using inaccurate scientific information in its metaphor. You would judge the truth or falsity of the poem on whether the actual claims it *was* making, it's real, underlying point, was true or false.