Yes.

Or at least, it was well-attested in the events that led to the award of the Medal of Honor to Private First Class Milton Lee Olive for actions on 22 October 1965.

https://www.cmohs.org/recipients/milton-l-olive-iii

PFC John A. Barnes, two years later.

https://www.cmohs.org/recipients/john-a-barnes-iii

For a more modern example, see Petty Officer Michael Monsoor.

https://www.cmohs.org/recipients/michael-a-monsoor

While this is far from an exhaustive list, altruistic suicide was and is uncommon, and there is a reason these people received the Congressional Medal of Honor. Throwing oneself on a grenade is occasionally survivable, Lance Corporal Matthew Croucher, Royal Marines, used his rucksack to absorb the blast from a grenade in 2008, and was in good enough shape to refuse evacuation and continue to fight. He was awarded the George Cross. Then-Lance Corporal Jacklyn Harris, USMC used his helmet and his body to cover a pair of grenades on Iwo Jima (at age 17!) and survived the subsequent blast, living until 2008. He was medically discharged from the Marines with 200 pieces of metal in his body, and was awarded the CMOH in 1945 . And of course, Corporal Kyle Carpenter, USMC (ret) also survived a grenade blast, having thrown himself on it, and is the youngest living recipient of the Medal of Honor.

So, yes it happened. The investigations from the Medal of Honor or the George Cross would be unlikely to come to completely fictional conclusions. It was hardly common or expected and most people wouldn't do it, or it wouldn't rate the highest decorations that exist.

As for the psychology behind it, you would have to dig up interviews with the few people who survived. I decline to speculate other than to say adrenaline is a hell of a drug.

The Vietnam War is outside the area that I normally study, so I'll apologize if I'm overstepping here. However, I think this question can be answered with some depth.

Yes, there were instances in Vietnam and in other conflicts where service members intentionally dove on grenades to protect others. Often, they were awarded a Congressional Medal of Honor. Mostly, they were awarded posthumously.

By searching through Medal of Honor citations for the Vietnam War (there are 263 listed), I've found 74 instances of soldiers, sailors, and Marines being awarded the Medal of Honor for intentionally falling on a grenade or other explosive device (mines, satchel charges, etc.), grabbing a grenade to throw it back, or in some cases, shielding a comrade's body from a grenade. Others are also cited for other actions in addition to diving on a grenade or shielding a squadmate's body. It's also likely that I've missed some by using the wrong search terms or another error I made. It's also possible that some service members, for whatever reason, weren't warded the Medal of Honor for doing something similar. In any event, you can search through the site linked above to read full citations.

I would hesitate to say it was "common" for servicemen to dive on grenades, but it certainly happened before Vietnam, in Vietnam, and since then. Here are the names of those I could find whose Medal of Honor citation reports that they fell on a grenade or other explosive, or used their body to shield others from a grenade:

(+denotes that they received their medal posthumously)

PFC James Anderson Jr., USMC+

SFC Richard Allen Anderson, USA+

SFC Webster Anderson, USA

PFC Oscar Palmer Austin, USMC+

SP4c John Philip Baca, USA

HC2c Donald Ballard, USN

LCpl Jedh Colby Barker, USMC+

PFC John Andrew Barnes III, USA+

SGT Ted Belcher, USA+

PFC Leslie Allen Bellrichard, USA+

SP4c Michael Reinert Blanchfield, USA+

SSG Hammett Lee Bowen, USA+

PFC Daniel Dean Bruce, USMC+

PFC Bruce Wayne Carter, USMC+

PFC Ronald Leroy Coker, USMC+

SSG Peter Spencer Connor, USMC+

LCpl Thomas Elbert Creek, USMC+

SP4c Larry Gilbert Dahl, USA+

Sgt Rodney Maxwell Davis, USMC+

LCpl Emilio Albert De La Garza Jr., USMC+

PFC Douglas Eugene Dickey, USMC+

SP4c Daniel O. Fernandez, USA+

SP4c Michael John Fitzmaurice, USA

SGT Charles Clinton Fleek, USA+

CPL Michael Fleming Folland, USA+

Sgt Paul Hellstrom Foster, USMC+

1LT Douglas Bernard Fournet, USA+

PFC James William Fous, USA+

SSG Robert Willard Hartsock, USA+

CPL Frank Rocco Fratellenico, USA+

SP4c Peter Mathew Guenette, USA+

SP4c Frank Aloysius Herda, USA

SFC Charles Ernest Hosking Jr., USA+

LCpl James Donnie Howe, USMC+

Sp4c James Allen Ingalls, USA+

PFC Robert H. Jenkins, USMC+

SP4c Donald Ray Johnston, USA+

PFC Ralph Henry Johnson, USA+

1LT Stephen Edward Karopczyz, USA+

CPL Terry Teruo Kawamura, USA+

SSgt Allan Jay Kellogg Jr., USMC

SP4c Thomas James Kinsman, USA

PFC Garfield McDonnell Langhorn, USA+

SP4c Robert David Law, USA+

2LT Robert Ronald Leisy, USA+

SGT Donald Russell Long, USA+

1LT Gary Lee Miller, USA+

SSG Frankie Zoly Molnar, USA+

PFC James Howard Monroe, USA+

SSG Robert Charles Murray, USA+

PFC David Paul Nash, USA+

PFC Milton Lee Olive III, USA+

SP4c Kenneth Lee Olson, USA+

SN David George Oullet, USN+

Cpl William Thomas Perkins Jr., USMC+

PFC Jimmy Wayne Phipps, USMC+

SGT Larry Stanley Pierce, USA+

CAPT Riley Leroy Pitts, USA+

PFC William David Port, USA+ (he survived but was captured and died later that year, before his medal was awarded)

SSG Lazslo Rabel, USA+

1LT Ronald Eric Ray, USA

SGT Anund Charles Roark, USA+

SP4c Hector Santiago-Colon, USA+

1LT Ruppert Leon Sargent, USA+

SSG Clifford Chester Sims, USA+

1LT Russell Albert Steindam, USA+

SGT Mitchell William Stout, USA+

SP4c Robert Francis Stryker, USA+

1LT John Earl Warren Jr., USA+

SP4c Dale Eugene Wayrynen, USA+

LCpl Roy Mitchell Wheat, USMC+

PFC Dewayne Thomas Williams, USMC+

PFC Alfred Mac Wilson, USMC+

LCpl Kenneth Lee Worley, USMC+

1SG Maximo Yabes, USA+