It certainly possible that someone will change their MAC address as part of an attack.
If a system is using the MAC address for identification (e.g., Wi-Fi access), then impersonating the MAC address may allow the attacker to impersonate a real user or at least gain basic access. Additional meaures such as 802.1X, NAC, etc. should be used in such cases.
Within a local network, the MAC address is used to direct network traffic, so impersonating the MAC address of a valid machine may be used in an attack, though it is often not necessary to for eavesdropping and Man in the middle attacks based on the attacker's location on the network.
Also, there are tools one can use to send out arbitrary data packets where the address is just spoofed in the packet.
Changing the MAC would not make a computer "invisible". If a system wants to be invisible, then it has to emit no network traffic. Window's machines by default are always sending out traffic if you just let them sit there, but other operating systems or configurations may not emit any automated network traffic.
If one changes his/her MAC address, and there is another system with the same MAC address, it may be difficult for systems to determine if the traffic is coming from the real or impersonated system. Note that for network security, you would want to test for other factors and fingerprints to distinguish if a system is legitimate, as well as possibly using a VPN, 802.1X, etc.
A more likely scenario is that an attacker will continually change his MAC address to make it more difficult to track activity over time if the MAC address is being used to correlate log activity.