Marco van Basten's side have been drawn in the toughest group at the tournament, with World Cup finalists Italy and France favourites to progress.
Bargkamp told the Dutch media: "Normally the Oranje take time to grow in a tournament, but with Italy and France as opponents you don't get that time. They must be ready from the start."
The Oranje will also be wary of the fourth team in the group, Romania, having finished second behind the eastern Europeans in qualification.
But despite the strength of their opposition in Austria and Switzerland, Bergkamp believes Holland have the potential to win through to the knockout stages.
"Technically the Dutch team look good. If they can start to get on a roll then the Oranje can go far."
Van Basten's side came in for some criticism during their qualification campaign when they twice failed to defeat Romania, drawing 0-0 at home before losing 1-0 away, while they could only beat minnows Luxembourg 1-0 in their two meetings.
A 2-1 defeat to Belarus in the Oranje's final match, when the team had already qualified, led to further grumblings among their critics.
And Bergkamp, who scored 37 goal in 79 appearance with his country to sit second on the Dutch all-time goalscorers list behind Patrick Kluivert, warned the team's tendency to blow hot and cold could cost them dear.
He said: "We can play extremely well, but not always. Many key players are still young, especially for the international work. Sometimes the stability is not present. When it works, it works well but when it doesn't it is really bad."