Nothing is stopping you from opening up Roth IRAs in different companies and
contributing the maximum to each one, but if you don't withdraw contributions
(and any earnings generated by the withdrawn contributions while within the
IRA) so as to leave a total contribution of $5500 for 2013, then Yes, you will be penalized. You do not report contributions to a Roth IRA on your tax
return, but companies report to the IRS on the IRA accounts they hold, and
the contributions and withdrawals/distributions made each year into each IRA
account. Each company will send
you a copy of the form that it sent in to the IRS so you won't be blind-sided
in this matter. You will then get a letter from the IRS informing you that you have to pay penalties (the technical term is
a 6% excise tax) for over-contributing to your IRA. If you obdurately refuse
to remove the excess contributions (and any earnings generated
by the excess contributions), that tax will be assessed in the years
following too, until the matter is set right..
Note also that the maximum allowable contribution is reduced (and ultimately eliminated entirely) for people with high Modified Adjusted Gross Income (MAGI).
Some people make a contribution to a Roth IRA each year but then withdraw it in
timely fashion when they discover that their MAGI is too high (often
during the preparation of their income tax returns).