On the same physical link, no. The source needs to finish sending a packet before it can start a new one. Packets are atomic and cannot overlap.
Physically, with the limited propagation speed of a signal, the destination may still be receiving the previous packet while the source has already started transmitting the next one. With a 10 Gbit/s link and a propagation speed of 200,000 km/s, a single bit is just 2 cm "long". For a minimum sized Ethernet frame that's just 13.5 m, so a longer cable could actually hold one or more entire frames. [ _figures corrected_ ]
On a logical link, consisting of multiple physical links, it is normal for the source to send many packets while the destination is still receiving previous packets. In the TCP transport-layer protocol, the _window size_ defines the amount of data that can be "in flight".