Will a frame with these dimensions: 53CM CTC,CTT 56CM AND THE TOP TUBE IS 21 3/4"" fit someone who is 5-5, 5-6?

Thx.

If your seat tube, measured from the center of the bottom bracket spindle to the center-line of the top tube is 53cm, then I would say it is a pretty good fit for you. However, how can the center to top measurement be 56cm? I am a bit confused. Also, why would you report seat tube measurement in metic and top tube measurement is imperial terms?

Generally, I also offer my online customers a stand over height and suggest that they compare that height to their inseam measurement(riding shoes on). If there is between one and two inches clearance between the tube and your crotch, the fit will probably be OK.

I'd say the CTT measurement is either incorrect, or goes to a point above the top tube to the top of the clamp.
21 3/4" equates to 55cm.
You'd likely be needing a short handlebar stem to make it work.

No, unless you have really long legs. The stand over height on that bike would probably be 31 inches-ish, which probably is too tall for your inseam.

I agree with EjustE, the other fellows raise good points too. Even taking into account your less than accurate measurements, that bike's too big. That top tube is typical of a 55 or 56cm frame. Most bikes with such a top tube would have a 31' ST.Over ht. total TT length, including a 110 to 120 step is way to long irrespective of your taste whever it's racey or not. A core size of 53cm is not recognized as at all standard to a man of 5"6". Folks have wild ranges in their preferred sizes at BF but since you asked so ....
IF the bike feels OK after a test ride, OK.... but after a spell on any bike... is just OK, good enough ?

Its probably a little long in the TT area for a 5'6" person. Surprisingly, standover heights dont change much as you go up in size from 19" to 24" frames.