They definitely can have something to do with each other. Anxiety and Depression linked to ADHD is a very common symptom and it's typical for patients who suffer from undiagnosed ADHD to seek medical assistance for the first two and then being diagnosed. Anxiety and depression can often be a co-morbidity (I hope I use the word right) of ADHD.
For me personally ADHD gives me anxiety due to the feelings of stress it causes and the depression comes from feeling unable to handle/cope with it.
Your description sums up my experience quite well. I have been on Concerta for 5 months now 54mg + 18mg at lunch). In the beginning I felt like I could do "Everything, everywhere, all at once".
Now it has mellowed out a lot and I can even say I feel more affected by my AD(H)D on meds. But I know that is because the medication is helping in some aspects and some aspects alone and doesn't fix everything.
I have a greater ability to focus on the things I actually need to focus on (with some deviation of course), and the best part is that it heightens my mood level. Instead of waking up and being at a -5 I feel like i am more baselined with the medication. A baseline that actually makes me want to do things instead of just feeling that everything is an unclimbable mountain.