IntelliJ show JavaDocs tooltip on mouse over

Up until IntelliJ version 11, no, not just by hovering over it. If the cursor is inside the method- or attribute name, then CTRL+Q will show the JavaDoc on *nix and Windows. On MacOSX, this is CTRL+J.

Quote: "No, the only way to see the full javadoc is to use Quick Doc (Ctrl-Q)." -- http://devnet.jetbrains.net/thread/121174

EDIT

Since IntelliJ 12.1, this is possible. See @ADNow's answer.

For IntelliJ 13, there is a checkbox in Editor's page in IDE Settings

EDIT: For IntelliJ 14, the option has been moved to Editor > General page. It's the last option in the "Other" group. (For Mac the option is under the menu "IntelliJ Idea" > "Preferences").

EDIT: For IntelliJ 16, it's the second-to-last option in Editor > General > Other.

EDIT: For IntelliJ Ultimate 2016.1, it's been moved to Editor > General > Code Completion.

EDIT: For IntelliJ Ultimate 2017.2, aka IntelliJ IDEA 2017.2.3, there are actually two options:

1. In Editor > General > Other (section) > Show quick documentation on mouse move - delay 500 ms

• Select this check box to show quick documentation for the symbol at caret. The quick documentation pop-up window appears after the specified delay.

1. In Editor > General > Code Completion (sub-item) > Autopopup documention in 1000 ms, for explicitly invoked completion

• Select this check box to have IntelliJ IDEA automatically show a pop-up window with the documentation for the class, method, or field currently highlighted in the lookup list. If this check box is not selected, use Ctrl+Q to show quick documentation for the element at caret.
• Quick documentation window will automatically pop up with the specified delay in those cases only, when code completion has been invoked explicitly. For the automatic code completion list, documentation window will only show up on pressing Ctrl+Q.

EDIT: For IntelliJ Ultimate 2020.3, the first option is now located under Editor > Code Editing > Quick Documentation > Show quick documentation on mouse move