But, in returning hero Wayne Rooney, the former Toffees skipper reckons they have the ideal man to shoulder that burden.
Dave Watson was the last Everton captain to lift a trophy 22 years ago when they beat Manchester United at Wembley to win the FA Cup.
The challenge of ending the Merseysiders’ long wait for silverware is unlikely to be a daunting one for Rooney who returned this month after winning 16 honours in 13 trophy-laden years at Manchester United.
Rooney’s signing might be a little out of keeping with the rest of boss Ronald Koeman’s summer business so far and the more youthful squad he is building at Goodison Park.
While Rooney is 31 Koeman’s five other signings are all no older than 24.
But Neville said: “The signing of Rooney gels it all together for Everton. There is no way Wayne Rooney would go back to Everton if he didn’t think they were going to win things, that is for sure.
“He is a winner and will bring that winning mentality to the changing room.
“What he will also bring, that I experienced a little bit at Everton, is that there is a little bit of fear of what it takes to actually win a trophy.
“Everton haven’t won a trophy since 1995 and it was always a noose around my neck when I was captain.
“It was something I felt really pressured into doing because you are always compared to those captains and the teams that were great.
“I think Rooney has big enough shoulders to spread that kind of experience in the dressing room.”
As well as re-signing Rooney, Koeman has made a real statement of intent in the transfer market this summer.
He has already splashed out over £80m on goalkeeper Jordan Pickford, defender Michael Keane, midfielder Davy Klaasen plus forwards Sandro Ramirez and Henry Onyekuru , who will spend the forthcoming season on loan at Anderlecht, with more arrivals certain to come.
The new faces have helped cushion the blow of Romelu Lukaku’s £75m departure for Manchester United.
Neville, speaking at the inaugural Star Sixes tournament at the O2, said: “If the transfer window closed today, Everton have done the best business.
“They have been very proactive, they have not waited for Lukaku to leave, they have said ‘let’s get our players in first and take some of the pressure off’.
“I remember when I was at Everton with David Moyes we sold Joleon Lescott one summer and Mikel Arteta towards the end and when you sell those players it puts added pressure on and fans get a little angry.
“I was concerned before they signed anyone as Tottenham bought in five or six players when they sold Gareth Bale and it didn’t quite work.
“But Everton have actually signed four or five players, the fans are happy, it softens the blow of Lukaku leaving with knowing full well there are one or two more to come.”
Koeman hopes his big-money spree will help Everton take another huge leap forward and crack the top six after finishing eight points off it last season.
And Neville reckons Tottenham, runners-up last season, have shown Everton how they can upset the order at the top of the table.
He said: ““The way they have spent their money and the young players they have coming through, they have to look at Tottenham.
“Four or five years ago Tottenham and Everton were on similar levels, Tottenham kicked on and Everton didn’t but now I think they are well capable of doing a Tottenham – producing a young team with a brilliant manager in Ronald Koeman.
“It looks like they have spent the money wisely.
“Everton have bought quality and what is more exciting is the young players breaking through – couple that with Wayne Rooney and it should be good.”