Although Reading lay in sixth place at the start of the day, Brighton overtook them by winning 2-1 away to Nottingham Forest, meaning the Royals' 2-2 draw at home to promoted Burnley was not enough to keep them in the play-off places.
However, Adkins feels his side will learn the lessons from their heartache.
"It's important you feel the pain and the hurt because that's what makes you work even harder," he said. "You need to have a negative experience to make you stronger.
"We can't change anything now, we've given it a good go, and the future is still very bright at this club. We'll take a couple of days to chill out and then reflect on it.
"This was effectively a play-off game and we started brightly. We changed shape in the second half and got a good stranglehold on the game.
"We were well aware what was happening at Nottingham. There were a lot of oohs and aahs going on. We were going for everything. We couldn't have done much more."
Needing to win against Burnley to definitely rubberstamp their place in the play-offs, Reading were rewarded in the 16th minute when they went ahead, albeit it with a large slice of good fortune.
Jobi McAnuff broke clear, after being released by a pass from Adam Le Fondre, and his low cross to the near post was turned into his own net by Kieran Trippier.
But Burnley, with only five defeats this season, were quick to respond, scoring twice in the space of eight minutes to take a 2-1 half-time lead.
Scott Arfield grabbed the leveller in the 20th minute and, with the Reading rearguard still in disarray, Burnley forged ahead when Danny Ings thundered in his 26th goal of the season from close range.
Ings was denied another soon after by a fine save from home keeper Alex McCarthy, but Reading grew steadily more dominant in the second half and equalised through a stunnign volley from Garath McCleary.
Try as they did, though, Reading could not add to it and were left at the mercy of Brighton's late winner at the City Ground.
Burnley captain Jason Shackell hacked a goalbound shot from Pavel Pogrebnyak off the goal-line and keeper Tom Heaton made fine saves from McCleary, Alex Pearce and Sean Morrison as Reading's luck finally ran out.
"That was just about a fair result," Burnley manager Sean Dyche said.
"Reading obviously had to go for it but we defended very well. After the own goal, I thought that we were good value.
"I was very happy with the team and individual performances. There were so many good things today, good habits and good displays. I'm proud of all the players and all the staff.
"We had to focus on just ourselves. That's the marvel of the last day of the season, with all its twists and turns. But Reading will come back. They're a good club."