yes u can..
but expect some noob response like if u do that, u won't be getting any promotion pamphlet together with your billing that might come next time, bla bla bla...

they cant even have e-statement 

this would require you to have budgeted for the current month statement before the statement is out, if my understanding is correct. and provided u get paid end of the month. i.e 6 oct statement already budgeted from sept pay (out end sept)
But this would mean you are vulnerable to any sudden spends between the budgeting and the statement date.

However it is true that all spend from 7th of the month only needs to be paid end of next month, if you do proper tracking and budgeting.