I am an independant contractor for two businesses. My expenses (cell phone, internet, etc) are used for both businesses. Can I claim the same expenses twice?

No you cannot claim the same deduction twice in its entirety. You will have to allocate expenses between two businesses depending on the usage. For example if you had $100 for cell phone expenses and you used 20% of the time for the first business, 30% of the time for second business, and 50% for personal usage, then you can take a deduction for $20 for the first business, $30 deduction for the second business and the rest $50 cannot be deducted since its for personal use. 

If you have an expense that is both personal and business, you can only deduct the amount of the expense that can be directly allocated to business use.  If audited, you must have some type of reliable method to determine the cost that you allocate to the business, otherwise the IRS can deny the expense.