I exercise my fretting hand with this for 10 minutes a day on my way to work. After work at night at about midnight, I practice guitar.

Am I overdoing it? I make sure I stretch my fingers before and after exercising or playing guitar. Should there be a day I don't do the hand exercises? Or stop playing guitar for a day as well?
10 mins a day is fine. Although you can overdo exercises and get tendonitis, just stop doing it if you feel pain though.
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its possible but 10 minutes probably wouldnt cause it, (guitar is known to cause aurtheritis) afterall so maybe thats the case? do they feel bruised or just overused?