I have had this happen a few times now.
Whenever i migrate a magento website to a new server the contact form always stops working.
Any ideas why this is breaking?
Have you upgraded the Magento version in the process? In Magento 1.8, "Form Keys" were introduced as a security measure. Having an old template and a new Magento can cause exactly that: some forms not submitting correctly, you won't get an error message but just return on the form page.
In this case, you can add the following to the PHTML file inside the <form> tags:
<?php echo Mage::getSingleton('core/session')->getFormKey() ?>