Currently, no. The receiver will reject an attempt for less than the requested amount and fail with incorrect_or_unknown_payment_details.
Each invoice has a payment_hash which a payment corresponds to, and the payment hash is generated from a random payment_preimage. This preimage must be unique for each payment, else it may be possible for any party along the route of a payment to take money without forwarding the payment on.
When the payee receives the full amount for a payment, they release the payment_preimage to the penultimate hop in the route, who then removes the HTLC from their channel and the payment completes. The penultimate hop then forwards the payment_preimage to the hop before them, and so on, until it reaches to payer.
You can see that, if only half of the payment was done, then every participant along the route would already have knowledge of the payment_preimage. If you attempted to pay again with the same payment_hash, then any hop in that route could instantly remove the HTLC and take the payment by using the payment_preimage they already know, without forwarding the payment to the next hop.
There is work in progress to allow payments to be broken up and send along multiple routes, with the payee only surrendering the payment_preimage if the full amount from all routes is received. This is known as AMP (Atomic Multi-path Payments). There are multiple proposals for this which have been discussed on the mailing list.
