This would be a deficiency (or so some say) of the X clipboard design. There is no "storage buffer" for clipboard data.
A process, when it hits Copy (SetClipboardData) registers that it owns the clipboard. When another program hits paste, only then is the data transfered between the processes. It is this way to support content negotiation... where the data that gets copied or pasted might be a different format depending on where you paste it. Plain text, HTML, RTF, etc.
A downside is when the owning program closes, the data is lost.