Can you increase the paste buffer size in IntelliJ IDEA

Yes

Go to:

File > Settings > Editor > Limits > Maximum number of contents to keep in clipboard

...and change the number.

Update for IntelliJ 14 and Android Studio 1.2
In new versions, it appears under a slightly different level in the menu...

Go to:
File > Settings > Editor > General
Then scroll down to find:
Limits > Maximum number of contents to keep in clipboard

If they decide to move it again, you can always just search for "max clipboard", that is:

Then type "max clipboard" in the dialog that comes up.

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For IntelliJ IDEA 2020.1 it can be done by edit properties in Registry. Goes to Help > Find Action and type Registry

registry

search for clipboard

Goes to File > Settings > Editor > General And then find Limits section

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