Ireland will head into the championships with few onlookers giving them a great deal of chance of beating two of Spain, Italy and Croatia to a quarter-final spot.
However, 36-year-old keeper Given, who was part of the team which very nearly reached the last eight at the 2002 World Cup finals in the Far East, is convinced he and his team-mates can make a genuine impact in Poland and Ukraine.
Asked if there was a confidence within the squad, he said: "Of course there is. I think there's a real belief as well that we are a good team as well.
"People outside the country might not think that, but as a group of players and the management and even the supporters, we are a tight-knit group and we have got to keep that going throughout the championships.
"We would like to go a long way."
That confidence increased markedly on Thursday when both Given and experienced defender John O'Shea returned to full training at the Republic's Borgo A Buggiano base in Tuscany after a week of debate over their fitness.
The Aston Villa man was so concerned about the knee he jarred while training in Malahide last week that he flew to consult a London specialist, and Trapattoni's fears over O'Shea were partly responsible for his decision to include Paul McShane rather than Kevin Foley in his final 23.
The repercussions of that saga were still being felt in Montecatini as a clearly riled Giovanni Trapattoni hit back in the wake of Foley's claim that he had been "betrayed" by the manager.
However, while Given, like the rest of the squad, had sympathy for the Wolves defender, he agreed with Trapattoni's assertion that the campaign is about far more than one man and his disappointment.
He said: "Kevin is gutted, as we are gutted for him as well. But as the manager said, it's not about one player, it's about the team and the nation.
"He [Trapattoni] felt it was the right decision and we have got to respect his decision.
"We are gutted for him, but I am sure he is back at home and he is going to support the team in the summer."
Full-back Stephen Kelly suggested after Tuesday night's 5-0 friendly victory over a Tuscan Select XI that the Foley issue might even help to strengthen resolve within the squad.
But Given is confident there will be no need for any extra motivation as the opening game against Croatia in Poznan on June 10 fast approaches.
He said: "We are pretty tight anyway, to be honest. I don't think it needs anything like that. We are all together, players and fans alike."