They ought to challenge and it's not entirely unachievable, but to manage it would be a big over-performance and a fair few stars would need to align. The question with regards to Arsene Wenger is, in my opinion, a question of whether you're just sick of the journey by now. Because I think the destination is likely to be much the same regardless of the manager. Similar issues would await him. He might fix some of our recurring issues, but it's likely that another set would emerge.
It's a fair, and I think, accurate comment. Chelsea and City have the ability to spend far more money than Arsenal; they are not relying on the Mathieu Flaminis and Olivier Girouds of this world to be long-term first choice options. Instead, Chelsea can spend £20m on Nemanja Matic in January, and all four of Manchester City's strikers have cost more money than Olivier Giroud, and all, aside from perhaps the perpetually injured Stevan Jovetic, would start for Arsenal.
As Stillman points out, if Arsenal are to win the league, everything has to work. The annual injury crisis, which is by no means an excuse, but is a reason, has to not happen; of course, Arsenal have to figure out ways to avoid it being so devastating. Even that, though, goes back to money. City lose Sergio Agüero to a hamstring injury, and Edin Džeko pops up to score two against Manchester United. Arsenal are relying on Yaya Sanogo and Chuba Akpom if Giroud gets injured, and yes, Arsenal should have signed a striker, but they a) thought that a deal had been agreed for Gonzalo Higuain, and when it was reneged on by Real Madrid, b) went for Luis Suarez, who had an attainable release clause. Liverpool have had an excellent season, yet most of that seems to be down to playing just one game a week; throw European competition and a deeper cup run, and their lack of depth probably sees them in a worse league position.
Arsenal have the third or fourth highest wage bill every season, and every season they finish third or fourth. It will likely happen again this season; the only difference has been that Arsenal were top for 3 months. So while it is fair to criticise Wenger for the collapse, he's the same manager that had them top longer than any other side thus far. While comments on Wenger's continued suitability to the job are fair, and while there will be serious questions and concerns should Arsenal fail to win this season's FA Cup, one has to remember that the same manager that is overseeing a run of bad form also oversaw a sustained run of excellent form that only ended when Arsenal lost Theo Walcott and Aaron Ramsey, and then Mesut Özil.
That isn't to say that Arsenal should be content with just third or fourth; Arsenal should be looking to improve as best they can with the resources that are available, and this summer must be gotten right. On the other hand, though, Wenger has had faith in players--Laurent Koscielny, Theo Walcott and Aaron Ramsey, to name some examples--that other managers wouldn't. Any success Arsenal are going to have will be down in a large part to those three. There's a core of players that Wenger has built that another manager may have given up on. Again, this isn't to excuse the utterly pathetic displays away at Liverpool or Chelsea this season, but Arsenal are in the position they are because of the manager: he's the same man, who not 3 months ago, was a genius for having Arsenal top. Somethings have to change, but for now, the manager does not.