Question: We are charged some amount of interest if we miss our credit card payments. Is this the same sin as, lending at or paying interest intentionally?
You would not be responsible for missing a credit card payment in case of financial impossibility. However, you must be utterly careful not to cause these interest-based transactions, by accounting for your income and expenses from the previous month.
If you do not make your credit card payment although you could, then it means you have payed interest. Paying interest is unlawful just as lending at interest. Our Prophet (p.b.u.h.) enjoined:
“May Allah curse the giver of interest, the receiver and the consumer of interest.” 
Allah forgives you if you repent, and correct yourself by staying away from interest from then on. He not only forgives but also transforms your previous sins to good deeds. Bukhari, Sahih, Sales and Trade 25; Muslim, Sahih, Musaqat, 105. Allah’s curse is His averting from His mercy.