If you're at an interview, and are explaining training failures which occurred in the distant past, you can probably smooth it over as a learning experience.
But the trick is getting the interview...computer screening of applications seems to lower the ranking of applications with checkride failures, and with enough failures probably disqualifies the application. This process often does not consider how long ago it happened, your level of experience at the time, or the nature of the checkride. Keep in mind that airlines, in addition to concerns about pilot skill and train-ability, are also concerned about liability and media perception in the event of an accident...a pilot with an extensive history of training failures is not going to look good in the news or to the jury.
Frankly it's no longer a very fair system to applicants, you can get filtered out for any of a number of things, some of which you have no idea how to train or prepare for, or are even aware of. In the old days if you had the right amount of experience and internal recommendations (which meant your previous co-workers considered you a good pilot and employee) you could get an interview. The interview might be hard with a low success rate (many were) but at least you got to show your stuff and interact with real people.
So...try not to fail any checkrides. In addition to the obvious (study hard) you need to consider some other factors...
1) Flight preparation. OK so you studied really well. But are you fully comfortable with the manuevers you'll need to demonstrate? Sometimes instructors can be under pressure from flight school management (or even the student himself) to sign off students to meet timelines or costs goals. Don't let this happen to you. In fact, in today's environment I would actually add one extra training flight a day or two before the checkride just to be extra sure.
2) Examiner: Regardless of what anyone says, some or many examiners do have "unofficial" failure quotas, ie they cannot pass every checkride applicant or they will come under FSDO scrutiny. This is ESPECIALLY true for CFI initial checkrides. Make sure you know the reputation of the examiner, and your local FSDO culture. Ideally the examiner will have a close relationship with your school or instructor, this will prevent him from busting you for no particularly good reason just because he thinks he has a quota to meet. Also if at all possible use the same examiner for all of your checkrides. This way you'll have a personal relationship with him, so much less likely to be used as quota cannon-fodder.
3) Checkride Sign-off: The instructor signing you off should be experienced with the local area and know the examiner personally. If you get signed off by "the new guy" he may not know all the nuances of the examiner in question.
4) Aircraft: Do not do a checkride in an aircraft you're not familiar with (I mean the actual N-number, not just the model). Review the maintenance logs in detail prior to the checkride and make sure you understand what's in them. I know a guy who failed a checkride because he couldn't find the seat adjustment lever (it was in a different location than the plane he was used to flying).
5) Environment: Don't do a checkride away from your normal training area. This commonly happens due to examiner availability, where students are told to fly the plane to another town to meet the examiner at another airport. Just insist on doing it local, and be sure you've flown in all local practice areas where an examiner might want you to go.