No.
(` is for stress, ' is for softness)
Е is pronounced as [йэ]
!if stressed!
1) in the beginning of a word: ем [йэм], ей [йэй]
2) after ь and ъ: подъезд [падй`эст], колье [кал'й`э]
if unstressed
e is pronounced as something between [йэ] and [йи]
1) in the beginning of a word: енот [йин`от], еда [йид`а]
2) after ь and ъ: ученье [уч`эн'йи], подъезжал [падйижж`ал]
BUT after consonants
1) if stressed
e is pronounced as [э] and the previous consonant goes soft:
место [м'`эста], нет [н'эт]
2) if unstressed
e is pronounced as something between [э] and [и] and the previous consonant goes soft:
тебя [т'иб'`а], меня [м'ин'`а],
there is also a sound which is similar to the last vowel of "water" and "bigger" - reduced [э], which you can hear in unstressed posisions after hard consonants, like this (I will type is as э, but actually it's shorter and weaker than э): солнце [с`онцэ], тише [тишэ]