If you're using the correct username and password for your Sandbox instance, these issues could be preventing you from logging in.
Login troubleshooting steps
Old Login Information is Saved
If your web browser has saved your login information from before the last Sandbox refresh, your login attempts will likely fail. Clear your browser's cache, cookies, and saved passwords, then restart your browser to make sure that all old login information is deleted.
If you're still not able to log in, continue with these troubleshooting steps.
The URL is incorrect
There are cases where the generic login URL for Sandboxes (http://test.salesforce.com) should specify the instance where your Sandbox resides. You can find the instance in the "Location" column found by navigating to Setup | Sandboxes or Data Management | Sandboxes.
If you continue to experience login issues, change the login URL for the Sandbox from http://test.salesforce.com to http://csX.salesforce.com (where: csX represents your Sandbox instance, like cs1, cs2, cs3).
If your Sandbox was last refreshed prior to December 6, 2014, login issues may be resulting from the "Sandbox refresh results in usernames being prefixed and old sandbox still accessible" known issue. If you suspect this to be the cause, please log a case with Salesforce Support to investigate.
Additional information about logins
Also note that your sandbox email addresses are appended with an example domain (@example.com) and the original @ sign is replaced with =. If you are resetting the password, you might want to get your email address changed first. In order to change email address, please work with your system administrator or log a case with support.