Of course you can't bid way higher on a proposal than what's on your profile. Also, I wouldn't double my rate overnight.
As someone else suggested (either on this thread or a similar one) you up your rate gradually as you get more jobs.
What I'm wondering is: if the rate tip constantly suggests $27 or something along those lines and I'm charging $15, could I charge bit more? A lot of people on this thread are pretty confident about rates. Maybe because I'm still quite new, I'm not so sure...