What is the most significant problem usually encountered when using the accelerated method of traditional procurement?

I think that using Accelerated Method of Procurement is an indicative of system failure somewhere. It is always prone to high degree of risks esp in following areas:

Procedural Compliance

Ensuring Quality of Goods and Services being procured

Transparency of the procurement process

Non-competitive prices

Conflict of Interest

etc etc

You will make mistakes yet you will achieve your goal, You will get bad quality products and maybe at a high price.