Yes, of course it is.
Bitcoin is money and 1 Bitcoin is currently worth about $1,400 USD. So, what you are really asking is “Is it possible to earn $1,400 in a month?”
Day Trading Bitcoin and other cryptocurrencies - This is just one of the most popular mechanisms to make bitcoin on a daily, weekly or monthly basis. There are even paid services like CoinData that analyze the market to let you know when a particular coin is likely to increase in value so you know when to buy in. I have a friend who trades using this and he is up 91% since April 1st so it is well worth the average $2 a day that this service costs.
Arbitrage - This is a bit more tricky but the opportunity is there. This is where you buy bitcoin at an exchange and sell it at a higher price at a different exchange. I'm still trying to figure out how to do this effectively.
Generating Ad Revenue - if you have some sort of blog or online following you can put ads on your site that only pay out in BTC. It would take a lot of traffic to earn 1 btc in a month but it is definitely possible. Check out anonymous ads to learn more about this.
Mine Bitcoin - I don't think that this is particularly profitable but if you have a mining rig that puts out enough computing power, you can generate 1 btc of profit in a month. Visit CoinWarz to learn how to best allocate your mining power for the most profitability.
Good luck and I would love to hear any feedback that you have on any of this. Thanks.