I don't think any skater has won the Hart -trophy when they've had the Vezina -winning goalie in that team aswell. Atleast i can't remember, might have happened when the scoring totals were insane and few teams were dominating.
It's not that the Vezina -goalie is 100% certainly the MVP (although, usually is) but it's that they eat votes out of each other. Hard to win that way.
If we are to analyze list of winners from the same team, we will get the tendency:
Generational player [Morenz, Shore, Richard, Howe, Beliveau, Lafleur, Plante]
and\or
Top-20 player (at least) at their position [Hainsworth, Thompson, Lach, Durnan, Sawchuk, Mikita, Hall, Clarke, Parent, Dryden].
The only exception is Herb Gardiner from 90 years ago (that was fourth season Hart Trophy was awarded and back then defenseman dominated in voting results, kinda like right now its mainly forward award).
I could see Ovechkin and Holtby winning, but not Kuznetsov\Backstrom and Holtby, for example. [
theoretically]
1926-27 Montreal Canadiens - [
Hart: Herb Gardiner] - [
Vezina: George Hainsworth]
1927-28 Montreal Canadiens - [
Hart: Howie Morenz] - [
Vezina: George Hainsworth]
1932-33 Boston Bruins - [
Hart: Eddie Shore] - [
Vezina: Tiny Thompson]
1935-36 Boston Bruins - [
Hart: Eddie Shore] - [
Vezina: Tiny Thompson]
1937-38 Boston Bruins - [
Hart: Eddie Shore] - [
Vezina: Tiny Thompson]
1944-45 Montreal Canadiens - [
Hart: Elmer Lach] - [
Vezina: Bill Durnan]
1946-47 Montreal Canadiens - [
Hart: Maurice Richard] - [
Vezina: Bill Durnan]
1951-52 Detroit Red Wings - [
Hart: Gordie Howe] - [
Vezina: Terry Sawchuk]
1952-53 Detroit Red Wings - [
Hart: Gordie Howe] - [
Vezina: Terry Sawchuk]
1955-56 Montreal Canadiens - [
Hart: Jean Beliveau] - [
Vezina: Jacques Plante]
1963-64 Montreal Canadiens - [
Hart: Jean Beliveau] - [
Vezina: Charlie Hodge]
1966-67 Chicago Blackhawks - [
Hart: Stan Mikita] - [
Vezina: Glenn Hall]
1974-75 Philadelphia Flyers - [
Hart: Bobby Clarke] - [
Vezina: Bernie Parent]
1976-77 Montreal Canadiens - [
Hart: Guy Lafleur] - [
Vezina: Ken Dryden]
1977-78 Montreal Canadiens - [
Hart: Guy Lafleur] - [
Vezina: Ken Dryden]
Goalies that won both:
1961-62 Montreal Canadiens - [Hart: Jacques Plante] - [Vezina: Jacques Plante]
1996-97 Buffalo Sabres - [Hart: Dominik Hasek] - [Vezina: Dominik Hasek]
1997-98 Buffalo Sabres - [Hart: Dominik Hasek] - [Vezina: Dominik Hasek]
2001-02 Montreal Canadiens - [Hart: Jose Theodore] - [Vezina: Jose Theodore]
2014-15 Montreal Canadiens - [Hart: Carey Price] - [Vezina: Carey Price]
Although, it is worth noting the following for Vezina Trophy:
When introduced, the Vezina Trophy was awarded to the top goaltender in the league. Several winners, including Hainsworth in 1927 and Gardiner in 1934, did not have the lowest goals-against average. In 1946, the NHL Governors further clarified the criteria for winning. It was to go to the team with the fewest goals scored against it during the season. The goalie playing the most games for the team was awarded the Trophy. In 1965, the NHL Governors began allowing teammates to share the Vezina Trophy. From 1965 to 1981, the Vezina was awarded to any goaltenders who played 25 or more games for the team allowing the fewest goals against.
The NHL adopted the current criteria for the Vezina Trophy beginning in 1981–82. The William M. Jennings Trophy was created as a new award for the goaltender(s) playing 25 or more games for the team allowing the fewest goals against.
So under last voting system it hasn't happened once. Most of it because of Gretzky\Lemieux\Hasek domination (14 Harts in 19 seasons) and non comparable (to old times) parity in salary cap era (last 11 seasons).