The British (with a lot of help from the Poles and the French) and later the Americans cracked Enigma and most of the important Japanese diplomatic and military cypher/codes. Later in WW2 they had the help of crude analog computers that speeded up the process.
They proceeded from the premise that there was human readable sensible information in those endless series of 4 or 6 letter groups. Their task was much facilitated by operator errors - sending the same message in different codes/ciphers, using the same code pages on subsequent days, repeated phrases like, "Your Excellency" and so on and so on. Given time and enough data all codes/ciphers can be cracked - except for proper "one-time-pad" codes.
But then, how random is random? I have dozens of ways of producing pseudorandom numbers (best to start with a hardware RNG and then subject it to cycles of PRNG). A method of generating (AND conveying it securely to the recipient) genuinely random numbers is not easy. Enforcing the correct use of these numbers is virtually impossible.
But whatever you do there MUST be entropy in the message - given enough messages, enough knowledge of your adversary, the type of data likely to be communicated and enough time (and speed increases daily) that entropy is theoretically discoverable.