Musnad al-Imam Ahmad is not a sahih book, as Imam Ahmad never claimed in his book to quote only sahih narrations, rather than collecting the biggest amount of sunnah which is used as evidence by scholars.
Ahmad himself said: "I have only included a hadith in this book if it had been used as evidence by some of the scholars."
And ibn al-Jawzi claimed that the Musnad contains ahadith that are fabricated.
Note that some scholars say that the son of Imam Ahmad added a couple of ahadith to his fathers Musnad and this was the source of many da'if ahadith in it.
Al-Haafiz al-‘Iraqi (may Allah have mercy on him) said:
With regard to the presence of da‘eef (weak) hadiths in it, that is certain; in fact it even contains some mawdoo‘ (fabricated) hadiths.
Imam adh-Dhahabi (may Allah have mercy on him) said:
It contains a few hadiths that may be fabricated, but they are like a drop in the ocean.
Also note because of the compilation based on the name of sahaba the Musnad includes many repetitions of ahadith which are as in other books based on different chains up to the same sahabi or of different sahaba.
The only books that really are considered as sahih (by sunni Muslims) are sahih al-Bukahri and sahih Muslim, as both did big efforts to to separate the wheat from the chaff and they therefore have the biggest amount of reliable sahih narrations. Also the Muwatta' -when it comes to the ahadith it includes- is a sahih book, as you will find almost all ahadith in any other compilation.
Any other book like the sunan of ibn Majah, abi Dawod or an-Nassai or Jami' at-Tirmdihi or even the sahihs of ibn Hebban or ibn Khozaymah don't exclusively contain sahih narrations or at least their standard for accepting a hadith is much lower than the one of the two sahihs. Note that al-Bukhari also claimed that only the "musnad" (the ones he compiled with full chain) ahadith are considered as sahih, as he like other scholars of his time didn't compile ahadith exclusively but added sayings of sahaba and tabi'in and others to his book.
For example see the table in this site which tries to count the amount of sahih hadith and ends up with about 4400 hadith, about 4300 out of them are in both sahihs.
Muwatta' Malik موطأ مالك has 71% hadiths in common with the two sahihs (513 out of 732)
Sunan an-Nasa'i سنن النسائي has 68% hadiths in common with them (3622 out of 5352)
Sunan abi Dawod أبو داود سنن has 47% hadiths in common with them (2033 out of 4326)
Jami' at-Tirmdihi جامع الترمذي has 46% hadiths in common with them (1702 out of 3735)
See also wikipedia EN AR
And this fatwa