Transferring your ROTH IRA account from one institution to another is not considered a contribution, therefore you would still be eligible to make your IRA contribution of $6500 for 2013 as long as you are 50 years older or older. If you are younger than 50 you can only contribute $5,500. The limits apply to both IRAs and ROTH IRAs. As already mentioned the following link provides guidelines. http://www.irs.gov/Retirement-Plans/Amount-of-Roth-IRA-Contributions-That-You-Can-Make-F
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A rollover of an IRA to another IRA does not count as a current or prior year contribution. You should still be eligible to contribute in 2013 your full eligible amount. If you are over the age of 50 and you meet the income eligibility requirements for a Roth IRA, then yes you can contribute $6500 to your Roth IRA, if you have not made any other contributions to that Roth IRA for the tax year 2013.
To be sure of the rules go to the IRS site for all eligibility details: