Answer:
Yes, it is very possible. When hamsters are teething, you have to make sure they have things to chew. If not, their teeth can cause tremendous pain and result as ingrown. I suggest you get him a hamster treat that's good for gnawing, such as those edible snack shacks you can buy. 
http://s3.amazonaws.com/tc-photos/11442/product/standard/517093.jpg
Or, you could just give him traditional chew sticks, they do the job as well as any other chew toy.
http://www.purrfectpets.org.uk/images/products/small_animals/hamster%20nibble%20sticks.jpg
It also helps a lot to have empty toilet paper rolls around in the cage. Instead of throwing them out, put a few in the cage with him. He'll appreciate the fun and satisfaction. An evident sign of teething and such is if he chews on the cage's bars or any item inside, like a food bowl. This is perfectly healthy, he just wants to chew. The nibble sticks and such are sold for a reason.
Good luck, and make sure your little buddy has something to chew for it's health. It's a necessity in it's growth.
{ 2 comments… read them below or add one }
One Month Old?? Sounds more like your hamster misses his MOMMY!!! Make sure that you have a small box with shavings or shredded newspaper for the hamster to go into…This will help with his feeling of security….Don't worry about the Chewing…this is natural…Just provide safe items for chewing… such as carrots, celery and a few tree branches (apple are best)..OR go to the Pet Store and get some chew items for hamsters……good luck
yes it is VERY possible lol my hamster was the same way. They are like babies lol they go through growing stages and teeth pains
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