A common problem with WordPress sites is slow page load time.
This can be caused by excessive use of plugins including ones that contribute no value to the usefulness of the web site. Many poorly designed plugins can call on resources outside of the sites web host. The performance of these external services can't be controlled. Many custom themes can further slow down page load.
Slow page load times will affect your ranking with Google and can dramatically impact your Search Engine Optimization efforts. Google will take page load time into account when ranking your site. The other problem with slow page load times is that visitors will not wait for a slow page to load and may leave your site before they ever see it.
WordPress is and open source project. This means that hackers can examine the code and look for vulnerabilities to exploit. The good thing about this is that good developers can find and fix these issues.
There also have been cases of malicious code being contributed to the WordPress project that do not get noticed until web sites have been hacked.
WordPress updates are frequent to address these security concerns. Keeping WordPress up-to-date to avoid security issues can be nearly impossible. Each WordPress installation must be updated one at a time.
WordPress does not have a built-in and automatic backup system. There are plugins that can help with this but they must be installed and managed on each site separately. Most of these backup solutions are a manual process you must execute on a regular basis.
WordPress stores some of the data in the web site file system and other data in the database. This makes commercial backup system less than ideal.
Web developers like WordPress because it's easy to develop a simple web site with limited technical knowledge. Of course that can mean people with limited technical knowledge can hang out a shingle and begin developing web sites.
The truth is that very few end users actually edit their own content. This is because the interface is designed by developers, for developers. If the site has any sophisticated feature, managing the site can become extremely difficult.
There are over 40,000 plugins for WordPress. This can mean the interface for each WordPress installation is different. This can make it difficult for an editor use these inconsistent features.