In my 17 years of teaching experience, I have noticed that each student can definitely improve their skill set each week in order to be a better singer! A large part of breath support is your natural muscle memory kicking in with enough repetition of vocal warmups. This means that without trying too hard, you will achieve noticeable improvements every week that you are warming up for at least 3-4 days.
In my humble opinion, the only exception to your question is if someone is truly tone deaf. This means that they cannot hear most pitches. Singing will be a challenge for them much as dancing is a challenge for someone who cannot feel the rhythm in their body. To get a feeling for your sense of pitch, you can record yourself and play back the recording. If at least some of the pitches are correct, you are not tone deaf. If the whole thing is off pitch, you are likely tone deaf. Hopefully, this info was helpful!