The number of files you can store in your Dropbox is only limited by the amount of online storage space in your Dropbox account. If you've run out of space in your personal account, consider upgrading.
Note: The performance of the Dropbox application may start to decline when you store above 300,000 files. At that point you may observe some slowness or unexpected behavior of some features. We're always working to optimize Dropbox to better handle accounts with a large amount of files. Be sure to download the latest version of the desktop app.
Additional information on file numbers and account performance
The performance of the Dropbox application can decline if you have more than 300,000 files. This is a soft limit and depends highly on the hardware specifications of the computer running the Dropbox application. Some customers are able to have more files without issue.
If you have a lot of files and you'd like to try to improve the performance of individual computers, you can use selective sync to select only certain files to sync to your computer. You'll still be able to access all of your files from dropbox.com or other devices linked to your account.
Note: This limit is related to the number of files, not to their size. Dropbox accounts with a large number of files may take longer to sync. A delay in syncing or a prolonged sync is not necessarily a sign of a problem. Adjusting your bandwidth settings may increase your sync speed.