Bob,
You have to determine what random is....The way I read that OP and others are doing (or want to do ) this on periodic basis to compare the case progress. Also there is a direct relationship between CASE number and applicant. Each case number is tied to the individual applicant.
Example: I assume we do not suggest entering people's random SSN numbers to find out when they receive their refund from IRS. Do we ?
I follow the USCIS msg which clearly states "Unauthorized access attempts or use in excess of documented authority may subject you to a fine and/or imprisonment in accordance with Title 18, USC, Section 1030 or administrative penalties or dismissal".
If there is an investigation for a given case, the people who do this and may be asked to produce the evidence of authorization or face further investigation. If you believe, this could help your N-400 application, then do it.
Also, usage and "monitoring" by the others (for not own) case could drain USCIS web resources...
On balance, I realize the likelihood of prosecution/legal complication is slim. But remember, N-400 application raises the bar for being "Good Moral Character". Just imagine, if you mention to IO during interview that you monitor the other people cases.
My 2 cents advice is that If you believe you can publicly state that to USCIS or Gov officals, you are monitoring the other people cases of US Gov agency site without authorization and you will BE OK, then keep doing. If in doubt, do not do it. As the others stated, the benefit is slim and plus USCIS publishes the data by which the cases are guided for priority (you also have different avenues to check your own case, complain about it and etc. I know is not ideal, but the other case monitoring is not effective and can not produce any benefits )
........ now that you are in trouble too