Case or Lead Assignment Rules fail to set the Owner

1. Be sure the Case or Lead Assignment Rule is active:

In Salesforce Classic

• Setup | Customize | Case (or Lead) | Assignment Rules

In Lightning Experience

• For Lead: Setup | Feature Settings | Marketing | Lead Assignment Rules
• For Case: Setup | Feature Settings | Service | Case Assignment Rules

See the "Active" checkbox and edit the Assignment to make it active (just one can be active at a time)

2. Check the criteria or formula you have for each rule component:

In Salesforce Classic

• Setup | Customize | Case (or Lead) | Assignment Rules

In Lightning Experience

• For Lead: Setup | Feature Settings | Marketing | Lead Assignment Rules
• For Case: Setup | Feature Settings | Service | Case Assignment Rules

Edit the different lines as needed, check logic, fields, values, etc.

3. In addition to having correct criteria or formula in an active Case or Lead Assignment Rule, you also need to apply those rules for actual use. This is done in your Case and Lead page layouts.

In Salesforce Classic

• Setup | Customize | Case (or Lead) | Page Layouts | Edit the layout

In Lightning Experience

• Setup | Object Manager | Case (or Lead) | Page Layouts | Click the Page Layout Name

Click on the Layout Properties button and select the checkboxes for the Case Assignment:

Note: To force Case assignment rules, select Default under Case Assignment Checkbox. This means end users won’t see any option to turn off case assignment when editing a case. Choose to include the 'Assign using active assignment rules' checkbox by selecting Show on edit page. If both options are chosen, the assignment checkbox is displayed and is checked by default. The default setting will force the assignment rule to run unless the checkbox is deployed and a User manually deselects the checkbox.
 

Click OK and click Save.

Note:

• Cases can also be assigned to the Default Case Owner if the assignment rules fail to locate an owner (the case doesn't meet the criteria of any active rule), this might give the appearance that the Case is being assigned to the incorrect owner. Change the Default Case Owner by customizing the Support Settings. 
• The behavior of an Assignment Rule runs differently since it depends on how the record is edited, whether through Inline Editing or the Edit button.
• If Case was created via Email-to-Case, the Owner field on the Routing Address should be blank for the Assignment rule to trigger. For more information you can check this article: Assignment Rule is not firing when Case is created through Email-to-Case
• If a case is created via predefined action then assignment rules wouldn't be triggered as the assign using active assignment rules checkbox wouldn't be displayed on the page, feel free to upvote this idea : https://ideas.salesforce.com/s/idea/a0B8W00000GdkJvUAJ/trigger-case-assignment-rules-when-using-a-publisher-action