There are many schools of thought like VisistAdvaita, Achintya Bheda Abheda and Dvaita which are followed by different sects of Vaishnavites. Most of these Vaishnava sects seem to be very critical of Advaita as they see it alien to their beliefs. However, I am a Vaishnava myself and attracted to Advaita Vedanta. Can I be both an Advaitin and Vaishnava?
Yes, Advaitha is too generic, you can be almost anything, when you are an Advaithi.
Besides, Adi Shankaracharya himself was a Vaishnava, he composed many stotrams on the highest Lord as Vishnu. And confusingly enough he was also a Shaktha, Shaiva etc.
Because one of the core philosophies of Advaitha is that the same ethereal Brahman underlies all existence and all forms. So yes, you can be an Advaithi and a Vaishnava.
Who says that one can't be a Advaitavadin & a Vaishnava together ?
Âdi Shankaracharya had written commentaries of Vishnu Sahasranâma & Bhagavadgîtâ.
Srîdhara Svami, the first commentator of the Srimad Bhâgavatam & Vishnu Purâna, was the disciple of an Advaita monk named Paramânanda Purî.
Madhusûdana Sarasvatî, a renowned Advaita scholar who composed a text named 'Advaitasiddhi' aline-by-line refutal of an Dvaita text named 'Nyayâmruta' composed by Madhva-scholar Vyasatirtha, who had close connections with the Vijayanagara emperors, along with other works defending Advaita philosophy from criticism like 'Advaitamanjarî', 'Advaitaratnarakshana'; had also composed commentaries on the Shandilya Bhaktisutra & Shivamahimna stotra along with works like 'Krishna kutuhala nataka', 'Bhakti sâmânya nirûpana', 'Hari lîlâ vyâkhyâ' & 'Ânandamandakinî' which exposes his deep devotion to Sri Krishna.
The animosity of almost all schools of Vaishnava theology towards Advaita philosophy arises from the fact that Ramanuja & Madhva had decried Shaiva doctrines as heresy & Adi Shankara, who is regarded to be an incarnation of Bhagavâna Shiva in Advaita circles, to be a crypto-Buddhist.
In order to be Vaishnava as well as an Advaitavâdin, it is better to study the Srimad Bhâgavatam along with the commentary of Sridhara Svami & the Bhagavadgîta along with the commentary of Madhusûdana Sarasvati.
Warakari sampradaya of Maharashtra, who are devotee of Vitthal ( Krishna ) follows advait philosophy with bhakti of krishna. They are biggest vaishnavas in Maharashtra and one of the biggest Vaishnava sect overall.
There are lot of Vaishnavas who are philosophically inclined towards advait nature of Paramtatva.