Q: Why is my Drupal site [insert_weird_symptom_here] ?
A: Probably because you've missed something easy.
I'm sure you did your due diligence and made sure everything looked good on the site report when the site went live, right? We can pretend so. Even if it was fine before, changes to the Drupal site or the server environment can cause unforeseen problems.
Your site report may not include alarms, bells, whistles, and whoopie lights... But it should.
One of the nice things about Drupal is its ability to report basic information about the operational status of the site. There's nothing fancy about the reporting but it's mature, widely-adopted system and an easy way to get eyes on the vitals of your site. There are two main pieces to the reporting--the site status report, and the watchdog event/error log.
As a best practice you should take a peek at the site report from time to time whether or not the site is experiencing problems. It's there for a reason and it can alert you to issues before they're issues. If something is already wrong, take a look and see if the status report indicates anything abnormal. If any of the modules on the site sense that something may not be quite right, the issue will likely be indicated here. By no means does the status report need to be clean at all times, but non-"green" items are worth a second look.
In my experience the most important items on the status report are the file permission related items and the indicator that the cron task has been running as expected. Drupal needs the ability to write files for various reasons (caching CSS and JS, uploaded image content, image styles, Boost cache, etc) and not being able to manage these files can make Drupal do some wonky things. Also when cron isn't running Drupal doesn't have a chance to expire its caches which cause unexpected behaviour. It's also going to have a negative impact on the operation of your site if you rely on cron to pull in feeds or dispatch emails. Also if you're using cron to manage backups, you could potentially be exposing yourself to the risk of losing data.
The watchdog log shows information on different events that happen during the operation of the site. Many entries are just notices that may provide information about various events but you may also see entries that indicate an issue with the server or even issues with the code. Be sure to thoroughly read the messages as they usually give a good idea as to what the problem indicates.
If it looks like a coding issue be sure to look into the issue. Many times it will be a bit of custom code that's encountering an unexpected condition. Even if the error doesn't indicate custom code, be sure to trace it back as an error with custom code may manifest as an error somewhere deep in the guts of a core function. If you're experiencing a coding error that is a legitimate issue with Drupal or or a contrib module, make sure you're running the recommended versions. If the error persists, file an issue and if you're capable please try to submit a fix.