In the present work, we discuss the glass-forming ability of various binary alloys in which the glassy phase was not formed even by melt spinning technique with high cooling rate of the melt up to 1 MK/s (some consisted of partly glassy phase), though by commonly accepted guidelines, these alloys could be as good glass-formers as many other binary glasses. The alloys studied belong to binary systems with multiple eutectics; the constituent elements have a negative enthalpy of mixing, and a significant variability of atomic size differences is observed from system to system. The results indicate the necessity of taking into account simultaneously various factors influencing the glass-forming ability including melt fragility.