But Hill-Wood was confident the Gunners would mount a serious title challenge this season and insisted the board were not worried about Arsene Wenger's failure to win a trophy for seven years.
The Arsenal chairman also said he was "disgusted" by those fans who abused manager Wenger last season as the club once again came up short.
Hill-Wood told the Evening Standard: "Arsene has money to spend but there's a limit. We can't spend £50million on one player.
"At a certain level, we can't compete. I don't think [majority shareholder] Stan Kroenke is going to put the sort of dollars in that [Roman] Abramovich or Sheikh Mansour are putting into Chelsea or Manchester City. That's not the way he thinks clubs should be run.
"Luckily, Arsene understands that. He got an economics degree from Strasbourg University so he's certainly no fool. He knows how a club should be run. That annoys a lot of people but clubs have to be sustainable. We're not going to go bankrupt in the way one or two other well-known clubs have.
"The Glasgow Rangers example is something we've all got to guard against. They spent far more money than they could afford. We're ambitious enough but we're not going to end in the same plight as Rangers. That is a fact of life. So my advice is, don't get miserable about it."
Last season saw Wenger come under more pressure than ever since he took charge 16 years ago, with some fans even heard to chant: "You don't know what you're doing".
Hill-Wood said: "What those fans shouted did not influence me one little bit. I was disgusted to hear them because Arsene's been absolutely outstanding. He is still outstanding."
This summer saw captain Robin van Persie become the latest big name to quit the club because he did not believe their ambitions matched his own.
Hill-Wood said: "We have a pretty good chance of challenging for the Premiership. I don't see why we cannot win it this year."
Arsenal's decline has been traced back by some to the departure of David Dein as vice-chairman.
Hill-Wood dismissed this, as well as any prospect of Dein being invited to return.
He also said he remained opposed to the club's second largest shareholder, Alisher Usmanov, being handed a place on the board.
Meanwhile, full-back Carl Jenkinson has claimed Sunday's win at Liverpool proved Arsenal do "mean business" this season.
The Gunners drew their opening two league games 0-0 but got three points on the board at Anfield, as well as keeping another clean sheet.
Jenkinson told their official website, www.arsenal.com: "After the two results we had, they weren't ideal and like I said before the game, if we win at Anfield, it would be a big statement of intent.
"We've gone and done that and shown we mean business.
"Maybe people were questioning us after the first two games, but we knew in ourselves that the goals would come and we've come to Anfield, which is no easy place to pick up three points, and we've won 2-0."
Arsenal's new-found defensive resilience has been credited to former defender Steve Bould, who was promoted to assistant manager during the summer.
Jenkinson said: "Steve has come in and he's been there and done it.
"He's always there giving us advice and tips, and we've had a long pre-season to work together and get our shape right.
"A lot of credit goes to the midfield, too.
"You need that shield in front of the back four and I think we're doing that fantastically at the moment and it's showing in our results."